Equals 0 in the quadratic equation. Quadratic equations. Solving complete quadratic equations. How to solve quadratic equations

We continue to study the topic “ solving equations". We have already met with linear equations and are moving on to get acquainted with quadratic equations.

First, we will analyze what a quadratic equation is, how it is written in general form, and give related definitions. After that, using examples, we will analyze in detail how incomplete quadratic equations are solved. Then we move on to solving the complete equations, obtain the formula for the roots, get acquainted with the discriminant of the quadratic equation and consider the solutions of typical examples. Finally, let's trace the relationship between roots and coefficients.

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What is a Quadratic Equation? Their types

First you need to clearly understand what a quadratic equation is. Therefore, it is logical to start talking about quadratic equations with the definition of a quadratic equation, as well as related definitions. After that, you can consider the main types quadratic equations: reduced and unreduced, as well as complete and incomplete equations.

Definition and examples of quadratic equations

Definition.

Quadratic equation Is an equation of the form a x 2 + b x + c = 0, where x is a variable, a, b and c are some numbers, and a is nonzero.

Let's say right away that quadratic equations are often called equations of the second degree. This is because the quadratic equation is algebraic equation second degree.

The sounded definition allows you to give examples of quadratic equations. So 2 x 2 + 6 x + 1 = 0, 0.2 x 2 + 2.5 x + 0.03 = 0, etc. Are quadratic equations.

Definition.

The numbers a, b and c are called coefficients of the quadratic equation a x 2 + b x + c = 0, and the coefficient a is called the first, or the highest, or the coefficient at x 2, b is the second coefficient, or the coefficient at x, and c is the free term.

For example, let's take a quadratic equation of the form 5x2 −2x3 = 0, here the leading coefficient is 5, the second coefficient is −2, and the intercept is −3. Note, when the coefficients b and / or c are negative, as in the example just given, then it is used short form writing a quadratic equation of the form 5 x 2 −2 x − 3 = 0, not 5 x 2 + (- 2) x + (- 3) = 0.

It is worth noting that when the coefficients a and / or b are equal to 1 or −1, then they are usually not explicitly present in the quadratic equation, which is due to the peculiarities of writing such. For example, in a quadratic equation y 2 −y + 3 = 0, the leading coefficient is one, and the coefficient at y is −1.

Reduced and unreduced quadratic equations

Reduced and non-reduced quadratic equations are distinguished depending on the value of the leading coefficient. Let us give the corresponding definitions.

Definition.

A quadratic equation in which the leading coefficient is 1 is called reduced quadratic equation... Otherwise the quadratic equation is unreduced.

According to this definition, quadratic equations x 2 −3 x + 1 = 0, x 2 −x − 2/3 = 0, etc. - given, in each of them the first coefficient is equal to one. And 5 x 2 −x − 1 = 0, etc. - unreduced quadratic equations, their leading coefficients are different from 1.

From any non-reduced quadratic equation by dividing both parts of it by the leading coefficient, you can go to the reduced one. This action is an equivalent transformation, that is, the reduced quadratic equation obtained in this way has the same roots as the original unreduced quadratic equation, or, like it, has no roots.

Let us analyze by example how the transition from an unreduced quadratic equation to a reduced one is performed.

Example.

From the equation 3 x 2 + 12 x − 7 = 0, go to the corresponding reduced quadratic equation.

Solution.

It is enough for us to divide both sides of the original equation by the leading factor 3, it is nonzero, so we can perform this action. We have (3 x 2 + 12 x − 7): 3 = 0: 3, which is the same, (3 x 2): 3+ (12 x): 3−7: 3 = 0, and beyond (3: 3) x 2 + (12: 3) x − 7: 3 = 0, whence. So we got the reduced quadratic equation, which is equivalent to the original one.

Answer:

Complete and incomplete quadratic equations

The definition of a quadratic equation contains the condition a ≠ 0. This condition is necessary for the equation a x 2 + b x + c = 0 to be exactly quadratic, since at a = 0 it actually becomes a linear equation of the form b x + c = 0.

As for the coefficients b and c, they can be zero, both separately and together. In these cases, the quadratic equation is called incomplete.

Definition.

The quadratic equation a x 2 + b x + c = 0 is called incomplete if at least one of the coefficients b, c is equal to zero.

In turn

Definition.

Full quadratic equation Is an equation in which all coefficients are nonzero.

Such names are not given by chance. This will become clear from the following considerations.

If the coefficient b is equal to zero, then the quadratic equation takes the form a x 2 + 0 x + c = 0, and it is equivalent to the equation a x 2 + c = 0. If c = 0, that is, the quadratic equation has the form a x 2 + b x + 0 = 0, then it can be rewritten as a x 2 + b x = 0. And with b = 0 and c = 0, we get the quadratic equation a x 2 = 0. The resulting equations differ from the full quadratic equation in that their left-hand sides do not contain either a term with variable x, or a free term, or both. Hence their name - incomplete quadratic equations.

So the equations x 2 + x + 1 = 0 and −2 x 2 −5 x + 0.2 = 0 are examples of complete quadratic equations, and x 2 = 0, −2 x 2 = 0.5 x 2 + 3 = 0, −x 2 −5 · x = 0 are incomplete quadratic equations.

Solving incomplete quadratic equations

From the information in the previous paragraph it follows that there is three kinds of incomplete quadratic equations:

• a · x 2 = 0, it corresponds to the coefficients b = 0 and c = 0;
• a x 2 + c = 0 when b = 0;
• and a x 2 + b x = 0 when c = 0.

Let us analyze in order how incomplete quadratic equations of each of these types are solved.

a x 2 = 0

Let's start by solving incomplete quadratic equations in which the coefficients b and c are equal to zero, that is, with equations of the form a · x 2 = 0. The equation a · x 2 = 0 is equivalent to the equation x 2 = 0, which is obtained from the original by dividing both parts of it by a nonzero number a. Obviously, the root of the equation x 2 = 0 is zero, since 0 2 = 0. This equation has no other roots, which is explained, indeed, for any nonzero number p, the inequality p 2> 0 holds, whence it follows that for p ≠ 0 the equality p 2 = 0 is never achieved.

So, the incomplete quadratic equation a · x 2 = 0 has a single root x = 0.

As an example, let us give the solution to the incomplete quadratic equation −4 · x 2 = 0. Equation x 2 = 0 is equivalent to it, its only root is x = 0, therefore, the original equation also has a unique root zero.

A short solution in this case can be formulated as follows:
−4 x 2 = 0,
x 2 = 0,
x = 0.

a x 2 + c = 0

Now let's consider how incomplete quadratic equations are solved, in which the coefficient b is zero, and c ≠ 0, that is, equations of the form a · x 2 + c = 0. We know that transferring a term from one side of the equation to another with the opposite sign, as well as dividing both sides of the equation by a nonzero number, give an equivalent equation. Therefore, it is possible to carry out the following equivalent transformations of the incomplete quadratic equation a x 2 + c = 0:

• move c to the right side, which gives the equation a x 2 = −c,
• and divide both of its parts by a, we get.

The resulting equation allows us to draw conclusions about its roots. Depending on the values ​​of a and c, the value of the expression can be negative (for example, if a = 1 and c = 2, then) or positive, (for example, if a = −2 and c = 6, then), it is not equal to zero , since by hypothesis c ≠ 0. Let us examine separately the cases and.

If, then the equation has no roots. This statement follows from the fact that the square of any number is a non-negative number. It follows from this that when, then for any number p the equality cannot be true.

If, then the situation with the roots of the equation is different. In this case, if you remember about, then the root of the equation immediately becomes obvious, it is a number, since. It is easy to guess that the number is also the root of the equation, indeed,. This equation has no other roots, which can be shown, for example, by the contradictory method. Let's do it.

Let us denote the roots of the equation just sounded as x 1 and −x 1. Suppose that the equation has one more root x 2, different from the indicated roots x 1 and −x 1. It is known that substitution of its roots in the equation instead of x turns the equation into a true numerical equality. For x 1 and −x 1 we have, and for x 2 we have. The properties of numerical equalities allow us to perform term-by-term subtraction of true numerical equalities, so subtracting the corresponding parts of the equalities gives x 1 2 −x 2 2 = 0. The properties of actions with numbers allow you to rewrite the resulting equality as (x 1 - x 2) · (x 1 + x 2) = 0. We know that the product of two numbers is zero if and only if at least one of them is zero. Therefore, it follows from the obtained equality that x 1 - x 2 = 0 and / or x 1 + x 2 = 0, which is the same, x 2 = x 1 and / or x 2 = −x 1. This is how we came to a contradiction, since at the beginning we said that the root of the equation x 2 is different from x 1 and −x 1. This proves that the equation has no roots other than and.

Let's summarize the information of this item. The incomplete quadratic equation a x 2 + c = 0 is equivalent to the equation that

• has no roots if,
• has two roots and if.

Consider examples of solving incomplete quadratic equations of the form a · x 2 + c = 0.

Let's start with the quadratic equation 9 x 2 + 7 = 0. After transferring the free term to the right side of the equation, it will take the form 9 · x 2 = −7. Dividing both sides of the resulting equation by 9, we arrive at. Since there is a negative number on the right side, this equation has no roots, therefore, the original incomplete quadratic equation 9 · x 2 + 7 = 0 has no roots.

Solve another incomplete quadratic equation −x 2 + 9 = 0. Move the nine to the right: −x 2 = −9. Now we divide both sides by −1, we get x 2 = 9. On the right side there is a positive number, from which we conclude that or. Then we write down the final answer: the incomplete quadratic equation −x 2 + 9 = 0 has two roots x = 3 or x = −3.

a x 2 + b x = 0

It remains to deal with the solution of the last type of incomplete quadratic equations for c = 0. Incomplete quadratic equations of the form a x 2 + b x = 0 allows you to solve factorization method... Obviously, we can, located on the left side of the equation, for which it is enough to factor out the common factor x. This allows us to pass from the original incomplete quadratic equation to an equivalent equation of the form x · (a · x + b) = 0. And this equation is equivalent to the combination of two equations x = 0 and a x + b = 0, the last of which is linear and has a root x = −b / a.

So, the incomplete quadratic equation a x 2 + b x = 0 has two roots x = 0 and x = −b / a.

To consolidate the material, we will analyze the solution of a specific example.

Example.

Solve the equation.

Solution.

Moving x out of parentheses gives the equation. It is equivalent to two equations x = 0 and. We solve the resulting linear equation:, and by dividing the mixed number by common fraction, we find. Therefore, the roots of the original equation are x = 0 and.

After gaining the necessary practice, the solutions to such equations can be written briefly:

Answer:

x = 0,.

Discriminant, the formula for the roots of a quadratic equation

There is a root formula for solving quadratic equations. Let's write down quadratic formula: , where D = b 2 −4 a c- so-called quadratic discriminant... The notation essentially means that.

It is useful to know how the root formula was obtained, and how it is applied when finding the roots of quadratic equations. Let's figure it out.

Derivation of the formula for the roots of a quadratic equation

Suppose we need to solve the quadratic equation a x 2 + b x + c = 0. Let's perform some equivalent transformations:

• We can divide both sides of this equation by a nonzero number a, as a result we get the reduced quadratic equation.
• Now select a complete square on its left side:. After that, the equation will take the form.
• At this stage, it is possible to carry out the transfer of the last two terms to the right-hand side with the opposite sign, we have.
• And we also transform the expression on the right side:.

As a result, we come to an equation that is equivalent to the original quadratic equation a x 2 + b x + c = 0.

We have already solved equations similar in form in the previous paragraphs when we analyzed them. This allows us to draw the following conclusions regarding the roots of the equation:

• if, then the equation has no real solutions;
• if, then the equation has the form, therefore, whence its only root is visible;
• if, then or, which is the same or, that is, the equation has two roots.

Thus, the presence or absence of the roots of the equation, and hence the original quadratic equation, depends on the sign of the expression on the right side. In turn, the sign of this expression is determined by the sign of the numerator, since the denominator 4 · a 2 is always positive, that is, the sign of the expression b 2 −4 · a · c. This expression b 2 −4 a c was called the discriminant of the quadratic equation and marked with the letter D... From this, the essence of the discriminant is clear - by its value and sign, it is concluded whether the quadratic equation has real roots, and if so, what is their number - one or two.

Returning to the equation, rewrite it using the discriminant notation:. And we draw conclusions:

• if D<0 , то это уравнение не имеет действительных корней;
• if D = 0, then this equation has a single root;
• finally, if D> 0, then the equation has two roots or, which by virtue can be rewritten in the form or, and after expanding and reducing the fractions to a common denominator, we obtain.

So we derived the formulas for the roots of a quadratic equation, they have the form, where the discriminant D is calculated by the formula D = b 2 −4 · a · c.

With their help, with a positive discriminant, you can calculate both real roots of the quadratic equation. When the discriminant is equal to zero, both formulas give the same root value corresponding to a unique solution to the quadratic equation. And with a negative discriminant, when trying to use the formula for the roots of a quadratic equation, we are faced with extracting the square root of a negative number, which takes us beyond the scope of the school curriculum. With a negative discriminant, the quadratic equation has no real roots, but has a pair complex conjugate roots, which can be found by the same root formulas obtained by us.

Algorithm for solving quadratic equations using root formulas

In practice, when solving quadratic equations, you can immediately use the root formula, with which you can calculate their values. But this is more about finding complex roots.

However, in the school course of algebra, it is usually not about complex, but about real roots of a quadratic equation. In this case, it is advisable to first find the discriminant before using the formulas for the roots of the quadratic equation, make sure that it is non-negative (otherwise, we can conclude that the equation has no real roots), and only after that calculate the values ​​of the roots.

The above reasoning allows us to write quadratic equation solver... To solve the quadratic equation a x 2 + b x + c = 0, you need:

• by the discriminant formula D = b 2 −4 · a · c calculate its value;
• conclude that the quadratic equation has no real roots if the discriminant is negative;
• calculate the only root of the equation by the formula if D = 0;
• find two real roots of a quadratic equation using the root formula if the discriminant is positive.

Here we just note that if the discriminant is equal to zero, the formula can also be used, it will give the same value as.

You can proceed to examples of using the algorithm for solving quadratic equations.

Examples of solving quadratic equations

Consider solutions to three quadratic equations with positive, negative and zero discriminants. Having dealt with their solution, by analogy it will be possible to solve any other quadratic equation. Let's start.

Example.

Find the roots of the equation x 2 + 2 x − 6 = 0.

Solution.

In this case, we have the following coefficients of the quadratic equation: a = 1, b = 2, and c = −6. According to the algorithm, first you need to calculate the discriminant, for this we substitute the indicated a, b and c into the discriminant formula, we have D = b 2 −4 a c = 2 2 −4 1 (−6) = 4 + 24 = 28... Since 28> 0, that is, the discriminant is greater than zero, then the quadratic equation has two real roots. We find them using the root formula, we get, here you can simplify the expressions obtained by doing factoring out the sign of the root with the subsequent reduction of the fraction:

Answer:

Let's move on to the next typical example.

Example.

Solve the quadratic equation −4x2 + 28x − 49 = 0.

Solution.

We start by finding the discriminant: D = 28 2 −4 (−4) (−49) = 784−784 = 0... Therefore, this quadratic equation has a single root, which we find as, that is,

Answer:

x = 3.5.

It remains to consider the solution of quadratic equations with negative discriminant.

Example.

Solve the equation 5 y 2 + 6 y + 2 = 0.

Solution.

Here are the coefficients of the quadratic equation: a = 5, b = 6 and c = 2. Substituting these values ​​into the discriminant formula, we have D = b 2 −4 a c = 6 2 −4 5 2 = 36−40 = −4... The discriminant is negative, therefore, this quadratic equation has no real roots.

If you need to indicate complex roots, then we apply the well-known formula for the roots of the quadratic equation, and perform actions with complex numbers :

Answer:

there are no real roots, complex roots are as follows:.

Note again that if the discriminant of a quadratic equation is negative, then at school they usually immediately write down an answer in which they indicate that there are no real roots, and complex roots are not found.

Root formula for even second coefficients

The formula for the roots of a quadratic equation, where D = b 2 −4 ln5 = 2 7 ln5). Let's take it out.

Let's say we need to solve a quadratic equation of the form a x 2 + 2 n x + c = 0. Let's find its roots using the formula we know. To do this, calculate the discriminant D = (2 n) 2 −4 a c = 4 n 2 −4 a c = 4 (n 2 −a c), and then we use the formula for roots:

Let us denote the expression n 2 - a · c as D 1 (sometimes it is denoted by D "). Then the formula for the roots of the considered quadratic equation with the second coefficient 2 n takes the form , where D 1 = n 2 - a · c.

It is easy to see that D = 4 · D 1, or D 1 = D / 4. In other words, D 1 is the fourth part of the discriminant. It is clear that the sign of D 1 is the same as the sign of D. That is, the sign of D 1 is also an indicator of the presence or absence of the roots of a quadratic equation.

So, to solve the quadratic equation with the second coefficient 2 n, you need

• Calculate D 1 = n 2 −a · c;
• If D 1<0 , то сделать вывод, что действительных корней нет;
• If D 1 = 0, then calculate the only root of the equation by the formula;
• If D 1> 0, then find two real roots by the formula.

Consider solving an example using the root formula obtained in this paragraph.

Example.

Solve the quadratic equation 5x2 −6x − 32 = 0.

Solution.

The second coefficient of this equation can be represented as 2 · (−3). That is, you can rewrite the original quadratic equation in the form 5 x 2 + 2 (−3) x − 32 = 0, here a = 5, n = −3 and c = −32, and calculate the fourth part of the discriminant: D 1 = n 2 −a c = (- 3) 2 −5 (−32) = 9 + 160 = 169... Since its value is positive, the equation has two real roots. Let's find them using the corresponding root formula:

Note that it was possible to use the usual formula for the roots of a quadratic equation, but in this case, more computational work would have to be done.

Answer:

Simplifying the View of Quadratic Equations

Sometimes, before embarking on the calculation of the roots of a quadratic equation by formulas, it does not hurt to ask the question: "Is it possible to simplify the form of this equation?" Agree that in terms of calculations it will be easier to solve the quadratic equation 11 · x 2 −4 · x − 6 = 0 than 1100 · x 2 −400 · x − 600 = 0.

Usually, a simplification of the form of a quadratic equation is achieved by multiplying or dividing both parts of it by some number. For example, in the previous paragraph, we managed to simplify the equation 1100x2 −400x − 600 = 0 by dividing both sides by 100.

A similar transformation is carried out with quadratic equations, the coefficients of which are not. In this case, both sides of the equation are usually divided by the absolute values ​​of its coefficients. For example, let's take the quadratic equation 12 x 2 −42 x + 48 = 0. the absolute values ​​of its coefficients: GCD (12, 42, 48) = GCD (GCD (12, 42), 48) = GCD (6, 48) = 6. Dividing both sides of the original quadratic equation by 6, we arrive at the equivalent quadratic equation 2 x 2 −7 x + 8 = 0.

And the multiplication of both sides of the quadratic equation is usually done to get rid of fractional coefficients. In this case, the multiplication is carried out by the denominators of its coefficients. For example, if both sides of the quadratic equation are multiplied by the LCM (6, 3, 1) = 6, then it will take on a simpler form x 2 + 4 x − 18 = 0.

In conclusion of this paragraph, we note that almost always get rid of the minus at the leading coefficient of the quadratic equation, changing the signs of all terms, which corresponds to multiplying (or dividing) both parts by −1. For example, usually from the quadratic equation −2x2 −3x + 7 = 0 one goes over to the solution 2x2 + 3x − 7 = 0.

Relationship between roots and coefficients of a quadratic equation

The formula for the roots of a quadratic equation expresses the roots of an equation in terms of its coefficients. Based on the root formula, you can get other dependencies between the roots and the coefficients.

The best known and most applicable formulas are from Vieta's theorem of the form and. In particular, for the given quadratic equation, the sum of the roots is equal to the second coefficient with the opposite sign, and the product of the roots is equal to the free term. For example, by the form of the quadratic equation 3 x 2 −7 x + 22 = 0, one can immediately say that the sum of its roots is 7/3, and the product of the roots is 22/3.

Using the already written formulas, you can get a number of other relationships between the roots and the coefficients of the quadratic equation. For example, you can express the sum of the squares of the roots of a quadratic equation through its coefficients:.

Bibliography.

• Algebra: study. for 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2008 .-- 271 p. : ill. - ISBN 978-5-09-019243-9.
• A. G. Mordkovich Algebra. 8th grade. At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., Erased. - M .: Mnemozina, 2009 .-- 215 p.: Ill. ISBN 978-5-346-01155-2.

Continuing the topic "Solving Equations", the material in this article will introduce you to quadratic equations.

Let's consider everything in detail: the essence and writing of the quadratic equation, we will set related terms, we will analyze the scheme for solving incomplete and complete equations, we will get acquainted with the formula of roots and the discriminant, we will establish connections between the roots and coefficients, and of course we will give a visual solution of practical examples.

Quadratic equation, its types

Definition 1

Quadratic equation Is an equation written as a x 2 + b x + c = 0, where x- variable, a, b and c- some numbers, while a is not zero.

Often, quadratic equations are also called second-degree equations, since in essence a quadratic equation is an algebraic equation of the second degree.

Let us give an example to illustrate the given definition: 9 · x 2 + 16 · x + 2 = 0; 7.5 x 2 + 3, 1 x + 0, 11 = 0, etc. Are quadratic equations.

Definition 2

The numbers a, b and c Are the coefficients of the quadratic equation a x 2 + b x + c = 0, while the coefficient a is called the first, or senior, or coefficient at x 2, b - the second coefficient, or the coefficient at x, a c called a free member.

For example, in a quadratic equation 6 x 2 - 2 x - 11 = 0 the highest coefficient is 6, the second coefficient is − 2 and the free term is − 11 ... Let us pay attention to the fact that when the coefficients b and / or c are negative, then a short notation of the form is used 6 x 2 - 2 x - 11 = 0, but not 6 x 2 + (- 2) x + (- 11) = 0.

Let us also clarify this aspect: if the coefficients a and / or b are equal 1 or − 1 , then they may not take explicit participation in the recording of the quadratic equation, which is explained by the peculiarities of the recording of the indicated numerical coefficients. For example, in a quadratic equation y 2 - y + 7 = 0 the highest coefficient is 1, and the second coefficient is − 1 .

Reduced and unreduced quadratic equations

According to the value of the first coefficient, quadratic equations are divided into reduced and non-reduced ones.

Definition 3

Reduced quadratic equation Is a quadratic equation, where the leading coefficient is 1. For other values ​​of the leading coefficient, the quadratic equation is not reduced.

Let's give examples: quadratic equations x 2 - 4 x + 3 = 0, x 2 - x - 4 5 = 0 are reduced, in each of which the leading coefficient is 1.

9 x 2 - x - 2 = 0- unreduced quadratic equation, where the first coefficient is different from 1 .

Any unreduced quadratic equation can be transformed into a reduced equation by dividing both parts by the first coefficient (equivalent transformation). The transformed equation will have the same roots as the given unreduced equation, or it will also have no roots at all.

Consideration of a specific example will allow us to clearly demonstrate the implementation of the transition from an unreduced quadratic equation to a reduced one.

Example 1

The equation is 6 x 2 + 18 x - 7 = 0 . It is necessary to convert the original equation to the reduced form.

Solution

According to the above scheme, we divide both sides of the original equation by the leading coefficient 6. Then we get: (6 x 2 + 18 x - 7): 3 = 0: 3 and this is the same as: (6 x 2): 3 + (18 x): 3 - 7: 3 = 0 and further: (6: 6) x 2 + (18: 6) x - 7: 6 = 0. Hence: x 2 + 3 x - 1 1 6 = 0. Thus, an equation is obtained that is equivalent to the given one.

Answer: x 2 + 3 x - 1 1 6 = 0.

Complete and incomplete quadratic equations

Let's turn to the definition of a quadratic equation. In it, we clarified that a ≠ 0... A similar condition is necessary for the equation a x 2 + b x + c = 0 was precisely square, since for a = 0 it essentially transforms into a linear equation b x + c = 0.

In the case when the coefficients b and c equal to zero (which is possible, both individually and jointly), the quadratic equation is called incomplete.

Definition 4

Incomplete Quadratic Equation Is such a quadratic equation a x 2 + b x + c = 0, where at least one of the coefficients b and c(or both) is zero.

Full quadratic equation- a quadratic equation in which all numerical coefficients are not equal to zero.

Let us discuss why the types of quadratic equations are given exactly such names.

For b = 0, the quadratic equation takes the form a x 2 + 0 x + c = 0 which is the same as a x 2 + c = 0... At c = 0 the quadratic equation is written as a x 2 + b x + 0 = 0 which is equivalent to a x 2 + b x = 0... At b = 0 and c = 0 the equation becomes a x 2 = 0... The equations that we obtained differ from the full quadratic equation in that their left-hand sides do not contain either a term with variable x, or a free term, or both at once. Actually, this fact gave the name to this type of equations - incomplete.

For example, x 2 + 3 x + 4 = 0 and - 7 x 2 - 2 x + 1, 3 = 0 are complete quadratic equations; x 2 = 0, - 5 x 2 = 0; 11 x 2 + 2 = 0, - x 2 - 6 x = 0 - incomplete quadratic equations.

Solving incomplete quadratic equations

The above definition makes it possible to distinguish the following types of incomplete quadratic equations:

• a x 2 = 0, such an equation corresponds to the coefficients b = 0 and c = 0;
• a x 2 + c = 0 at b = 0;
• a x 2 + b x = 0 at c = 0.

Let us consider sequentially the solution of each type of incomplete quadratic equation.

Solution of the equation a x 2 = 0

As already indicated above, such an equation corresponds to the coefficients b and c equal to zero. The equation a x 2 = 0 can be transformed into an equivalent equation x 2 = 0, which we get by dividing both sides of the original equation by the number a not equal to zero. It is an obvious fact that the root of the equation x 2 = 0 it is zero because 0 2 = 0 ... This equation has no other roots, which can be explained by the properties of the degree: for any number p, not equal to zero, the inequality is true p 2> 0, from which it follows that for p ≠ 0 equality p 2 = 0 will never be achieved.

Definition 5

Thus, for an incomplete quadratic equation a x 2 = 0, there is a unique root x = 0.

Example 2

For example, let's solve an incomplete quadratic equation - 3 x 2 = 0... It is equivalent to the equation x 2 = 0, its only root is x = 0, then the original equation also has a single root - zero.

Briefly, the solution is formalized as follows:

- 3 x 2 = 0, x 2 = 0, x = 0.

Solution of the equation a x 2 + c = 0

The next step is the solution of incomplete quadratic equations, where b = 0, c ≠ 0, that is, equations of the form a x 2 + c = 0... We transform this equation by transferring the term from one side of the equation to another, changing the sign to the opposite and dividing both sides of the equation by a number that is not equal to zero:

• carry over c to the right, which gives the equation a x 2 = - c;
• we divide both sides of the equation by a, we get as a result x = - c a.

Our transformations are equivalent, respectively, the resulting equation is also equivalent to the original one, and this fact makes it possible to draw a conclusion about the roots of the equation. From what the values ​​are a and c the value of the expression - c a depends: it can have a minus sign (for example, if a = 1 and c = 2, then - c a = - 2 1 = - 2) or a plus sign (for example, if a = - 2 and c = 6, then - c a = - 6 - 2 = 3); it is not zero because c ≠ 0... Let us dwell in more detail on situations when - c a< 0 и - c a > 0 .

In the case when - c a< 0 , уравнение x 2 = - c a не будет иметь корней. Утверждая это, мы опираемся на то, что квадратом любого числа является число неотрицательное. Из сказанного следует, что при - c a < 0 ни для какого числа p the equality p 2 = - c a cannot be true.

Everything is different when - c a> 0: remember the square root, and it becomes obvious that the root of the equation x 2 = - c a will be the number - c a, since - c a 2 = - c a. It is easy to understand that the number - - c a is also the root of the equation x 2 = - c a: indeed, - - c a 2 = - c a.

The equation will have no other roots. We can demonstrate this using contradictory method. To begin with, we define the notation for the roots found above as x 1 and - x 1... Let us assume that the equation x 2 = - c a also has a root x 2 which is different from the roots x 1 and - x 1... We know that by substituting in the equation instead of x its roots, we transform the equation into a fair numerical equality.

For x 1 and - x 1 we write: x 1 2 = - c a, and for x 2- x 2 2 = - c a. Based on the properties of numerical equalities, we subtract one true equality from the other term by term, which will give us: x 1 2 - x 2 2 = 0... We use the properties of actions on numbers to rewrite the last equality as (x 1 - x 2) (x 1 + x 2) = 0... It is known that the product of two numbers is zero if and only if at least one of the numbers is zero. From what has been said it follows that x 1 - x 2 = 0 and / or x 1 + x 2 = 0 which is the same x 2 = x 1 and / or x 2 = - x 1... An obvious contradiction arose, because at first it was agreed that the root of the equation x 2 differs from x 1 and - x 1... So, we proved that the equation has no other roots, except for x = - c a and x = - - c a.

We summarize all the reasoning above.

Definition 6

Incomplete Quadratic Equation a x 2 + c = 0 is equivalent to the equation x 2 = - c a, which:

• will have no roots for - c a< 0 ;
• will have two roots x = - c a and x = - - c a for - c a> 0.

Let us give examples of solving the equations a x 2 + c = 0.

Example 3

Quadratic equation given 9 x 2 + 7 = 0. It is necessary to find a solution to it.

Solution

We transfer the free term to the right side of the equation, then the equation will take the form 9 x 2 = - 7.
We divide both sides of the resulting equation by 9 , we arrive at x 2 = - 7 9. On the right side, we see a number with a minus sign, which means: the given equation has no roots. Then the original incomplete quadratic equation 9 x 2 + 7 = 0 will have no roots.

Answer: the equation 9 x 2 + 7 = 0 has no roots.

Example 4

It is necessary to solve the equation - x 2 + 36 = 0.

Solution

Move 36 to the right side: - x 2 = - 36.
Let's divide both parts into − 1 , we get x 2 = 36... On the right side there is a positive number, from which we can conclude that x = 36 or x = - 36.
Let's extract the root and write down the final result: an incomplete quadratic equation - x 2 + 36 = 0 has two roots x = 6 or x = - 6.

Answer: x = 6 or x = - 6.

Solution of the equation a x 2 + b x = 0

Let us analyze the third kind of incomplete quadratic equations, when c = 0... To find a solution to an incomplete quadratic equation a x 2 + b x = 0, we will use the factorization method. We factor out the polynomial on the left side of the equation, taking out the common factor outside the brackets x... This step will make it possible to convert the original incomplete quadratic equation to its equivalent x (a x + b) = 0... And this equation, in turn, is equivalent to a set of equations x = 0 and a x + b = 0... The equation a x + b = 0 linear, and its root is: x = - b a.

Definition 7

Thus, the incomplete quadratic equation a x 2 + b x = 0 will have two roots x = 0 and x = - b a.

Let's fix the material with an example.

Example 5

It is necessary to find a solution to the equation 2 3 x 2 - 2 2 7 x = 0.

Solution

Take out x brackets and get the equation x · 2 3 · x - 2 2 7 = 0. This equation is equivalent to the equations x = 0 and 2 3 x - 2 2 7 = 0. Now you need to solve the resulting linear equation: 2 3 · x = 2 2 7, x = 2 2 7 2 3.

We briefly write the solution to the equation as follows:

2 3 x 2 - 2 2 7 x = 0 x 2 3 x - 2 2 7 = 0

x = 0 or 2 3 x - 2 2 7 = 0

x = 0 or x = 3 3 7

Answer: x = 0, x = 3 3 7.

Discriminant, the formula for the roots of a quadratic equation

To find a solution to quadratic equations, there is a root formula:

Definition 8

x = - b ± D 2 a, where D = b 2 - 4 a c- the so-called discriminant of the quadratic equation.

The notation x = - b ± D 2 · a essentially means that x 1 = - b + D 2 · a, x 2 = - b - D 2 · a.

It will be useful to understand how the indicated formula was derived and how to apply it.

Derivation of the formula for the roots of a quadratic equation

Let us face the task of solving a quadratic equation a x 2 + b x + c = 0... Let's carry out a number of equivalent transformations:

• divide both sides of the equation by the number a, nonzero, we obtain the reduced quadratic equation: x 2 + b a · x + c a = 0;
• select the full square on the left side of the resulting equation:
  x 2 + ba x + ca = x 2 + 2 b 2 a x + b 2 a 2 - b 2 a 2 + ca = = x + b 2 a 2 - b 2 a 2 + ca
  After this, the equation will take the form: x + b 2 · a 2 - b 2 · a 2 + c a = 0;
• now it is possible to transfer the last two terms to the right-hand side by changing the sign to the opposite, after which we get: x + b 2 · a 2 = b 2 · a 2 - c a;
• finally, we transform the expression written on the right side of the last equality:
  b 2 a 2 - c a = b 2 4 a 2 - c a = b 2 4 a 2 - 4 a c 4 a 2 = b 2 - 4 a c 4 a 2.

Thus, we have come to the equation x + b 2 a 2 = b 2 - 4 a c 4 a 2, which is equivalent to the original equation a x 2 + b x + c = 0.

We analyzed the solution of such equations in the previous paragraphs (solution of incomplete quadratic equations). The experience already gained makes it possible to draw a conclusion regarding the roots of the equation x + b 2 a 2 = b 2 - 4 a c 4 a 2:

• at b 2 - 4 a c 4 a 2< 0 уравнение не имеет действительных решений;
• for b 2 - 4 a c 4 a 2 = 0 the equation has the form x + b 2 a 2 = 0, then x + b 2 a = 0.

Hence, the only root x = - b 2 · a is obvious;

• for b 2 - 4 a c 4 a 2> 0 it will be true: x + b 2 a = b 2 - 4 a c 4 a 2 or x = b 2 a - b 2 - 4 a c 4 a 2, which is the same as x + - b 2 a = b 2 - 4 a c 4 a 2 or x = - b 2 a - b 2 - 4 a c 4 a 2, i.e. the equation has two roots.

It is possible to conclude that the presence or absence of roots of the equation x + b 2 a 2 = b 2 - 4 a c 4 a 2 (and hence the original equation) depends on the sign of the expression b 2 - 4 a c 4 · A 2 written on the right side. And the sign of this expression is set by the sign of the numerator, (denominator 4 a 2 will always be positive), that is, by the sign of the expression b 2 - 4 a c... This expression b 2 - 4 a c the name is given - the discriminant of the quadratic equation and the letter D is defined as its designation. Here you can write down the essence of the discriminant - by its value and sign, it is concluded whether the quadratic equation will have real roots, and, if so, what is the number of roots - one or two.

Let's return to the equation x + b 2 a 2 = b 2 - 4 a c 4 a 2. We rewrite it using the notation for the discriminant: x + b 2 · a 2 = D 4 · a 2.

Let us formulate the conclusions again:

Definition 9

• at D< 0 the equation has no real roots;
• at D = 0 the equation has a single root x = - b 2 · a;
• at D> 0 the equation has two roots: x = - b 2 a + D 4 a 2 or x = - b 2 a - D 4 a 2. Based on the properties of radicals, these roots can be written as: x = - b 2 a + D 2 a or - b 2 a - D 2 a. And, when we open the modules and bring the fractions to a common denominator, we get: x = - b + D 2 · a, x = - b - D 2 · a.

So, the result of our reasoning was the derivation of the formula for the roots of the quadratic equation:

x = - b + D 2 a, x = - b - D 2 a, the discriminant D calculated by the formula D = b 2 - 4 a c.

These formulas make it possible, with a discriminant greater than zero, to determine both real roots. When the discriminant is zero, applying both formulas will give the same root as the only solution to the quadratic equation. In the case where the discriminant is negative, trying to use the square root formula, we will be faced with the need to extract the square root of a negative number, which will take us beyond the real numbers. With a negative discriminant, the quadratic equation will not have real roots, but a pair of complex conjugate roots is possible, determined by the same root formulas we obtained.

Algorithm for solving quadratic equations using root formulas

It is possible to solve the quadratic equation by immediately using the root formula, but basically this is done when it is necessary to find complex roots.

In the bulk of cases, it is usually meant to search not for complex, but for real roots of a quadratic equation. Then it is optimal, before using the formulas for the roots of the quadratic equation, first determine the discriminant and make sure that it is not negative (otherwise, we will conclude that the equation has no real roots), and then proceed to calculate the values ​​of the roots.

The reasoning above makes it possible to formulate an algorithm for solving a quadratic equation.

Definition 10

To solve a quadratic equation a x 2 + b x + c = 0, necessary:

• according to the formula D = b 2 - 4 a c find the value of the discriminant;
• at D< 0 сделать вывод об отсутствии у квадратного уравнения действительных корней;
• for D = 0, find the only root of the equation by the formula x = - b 2 · a;
• for D> 0, determine two real roots of the quadratic equation by the formula x = - b ± D 2 · a.

Note that when the discriminant is zero, you can use the formula x = - b ± D 2 · a, it will give the same result as the formula x = - b 2 · a.

Let's look at some examples.

Examples of solving quadratic equations

Let us give a solution of examples for different meanings discriminant.

Example 6

It is necessary to find the roots of the equation x 2 + 2 x - 6 = 0.

Solution

We write down the numerical coefficients of the quadratic equation: a = 1, b = 2 and c = - 6... Next, we act according to the algorithm, i.e. let's start calculating the discriminant, for which we substitute the coefficients a, b and c into the discriminant formula: D = b 2 - 4 a c = 2 2 - 4 1 (- 6) = 4 + 24 = 28.

So, we got D> 0, which means that the original equation will have two real roots.
To find them, we use the root formula x = - b ± D 2 · a and, substituting the corresponding values, we get: x = - 2 ± 28 2 · 1. Let us simplify the resulting expression by taking the factor outside the root sign and then reducing the fraction:

x = - 2 ± 2 7 2

x = - 2 + 2 7 2 or x = - 2 - 2 7 2

x = - 1 + 7 or x = - 1 - 7

Answer: x = - 1 + 7, x = - 1 - 7.

Example 7

It is necessary to solve the quadratic equation - 4 x 2 + 28 x - 49 = 0.

Solution

Let's define the discriminant: D = 28 2 - 4 (- 4) (- 49) = 784 - 784 = 0... With this value of the discriminant, the original equation will have only one root, determined by the formula x = - b 2 · a.

x = - 28 2 (- 4) x = 3, 5

Answer: x = 3, 5.

Example 8

It is necessary to solve the equation 5 y 2 + 6 y + 2 = 0

Solution

The numerical coefficients of this equation will be: a = 5, b = 6 and c = 2. We use these values ​​to find the discriminant: D = b 2 - 4 · a · c = 6 2 - 4 · 5 · 2 = 36 - 40 = - 4. The calculated discriminant is negative, so the original quadratic equation has no real roots.

In the case when the task is to indicate complex roots, we apply the formula for the roots, performing actions with complex numbers:

x = - 6 ± - 4 2 5,

x = - 6 + 2 i 10 or x = - 6 - 2 i 10,

x = - 3 5 + 1 5 · i or x = - 3 5 - 1 5 · i.

Answer: no valid roots; the complex roots are as follows: - 3 5 + 1 5 · i, - 3 5 - 1 5 · i.

In the school curriculum, as a standard, there is no requirement to look for complex roots, therefore, if during the solution the discriminant is determined as negative, the answer is immediately recorded that there are no real roots.

Root formula for even second coefficients

The formula for roots x = - b ± D 2 a (D = b 2 - 4 a n, for example 2 3 or 14 ln 5 = 2 7 ln 5). Let us show how this formula is derived.

Suppose we are faced with the task of finding a solution to the quadratic equation a · x 2 + 2 · n · x + c = 0. We act according to the algorithm: we determine the discriminant D = (2 n) 2 - 4 a c = 4 n 2 - 4 a c = 4 (n 2 - a c), and then use the root formula:

x = - 2 n ± D 2 a, x = - 2 n ± 4 n 2 - a c 2 a, x = - 2 n ± 2 n 2 - a c 2 a, x = - n ± n 2 - a ca.

Let the expression n 2 - a · c be denoted as D 1 (sometimes it is denoted by D "). Then the formula for the roots of the considered quadratic equation with the second coefficient 2 n will take the form:

x = - n ± D 1 a, where D 1 = n 2 - a · c.

It is easy to see that D = 4 · D 1, or D 1 = D 4. In other words, D 1 is a quarter of the discriminant. Obviously, the sign of D 1 is the same as the sign of D, which means that the sign of D 1 can also serve as an indicator of the presence or absence of roots of a quadratic equation.

Definition 11

Thus, to find a solution to the quadratic equation with the second coefficient 2 n, it is necessary:

• find D 1 = n 2 - a · c;
• at D 1< 0 сделать вывод, что действительных корней нет;
• when D 1 = 0, determine the only root of the equation by the formula x = - n a;
• for D 1> 0, determine two real roots by the formula x = - n ± D 1 a.

Example 9

It is necessary to solve the quadratic equation 5 x 2 - 6 x - 32 = 0.

Solution

The second coefficient of the given equation can be represented as 2 · (- 3). Then we rewrite the given quadratic equation as 5 x 2 + 2 (- 3) x - 32 = 0, where a = 5, n = - 3 and c = - 32.

We calculate the fourth part of the discriminant: D 1 = n 2 - ac = (- 3) 2 - 5 (- 32) = 9 + 160 = 169. The resulting value is positive, which means that the equation has two real roots. Let's define them according to the corresponding root formula:

x = - n ± D 1 a, x = - - 3 ± 169 5, x = 3 ± 13 5,

x = 3 + 13 5 or x = 3 - 13 5

x = 3 1 5 or x = - 2

It would be possible to carry out calculations using the usual formula for the roots of a quadratic equation, but in this case the solution would be more cumbersome.

Answer: x = 3 1 5 or x = - 2.

Simplifying the View of Quadratic Equations

Sometimes it is possible to optimize the form of the original equation, which will simplify the process of calculating the roots.

For example, the quadratic equation 12 x 2 - 4 x - 7 = 0 is clearly more convenient for solving than 1200 x 2 - 400 x - 700 = 0.

More often, the simplification of the form of a quadratic equation is performed by multiplying or dividing both parts of it by a certain number. For example, above we showed a simplified representation of the equation 1200 x 2 - 400 x - 700 = 0, obtained by dividing both parts of it by 100.

Such a transformation is possible when the coefficients of the quadratic equation are not coprime numbers. Then, usually, both sides of the equation are divided by the greatest common divisor of the absolute values ​​of its coefficients.

As an example, use the quadratic equation 12 x 2 - 42 x + 48 = 0. Determine the gcd of the absolute values ​​of its coefficients: gcd (12, 42, 48) = gcd (gcd (12, 42), 48) = gcd (6, 48) = 6. We divide both sides of the original quadratic equation by 6 and get the equivalent quadratic equation 2 x 2 - 7 x + 8 = 0.

By multiplying both sides of a quadratic equation, you usually get rid of fractional coefficients. In this case, multiply by the smallest common multiple of the denominators of its coefficients. For example, if each part of the quadratic equation 1 6 x 2 + 2 3 x - 3 = 0 is multiplied with the LCM (6, 3, 1) = 6, then it will become written in a simpler form x 2 + 4 x - 18 = 0.

Finally, we note that almost always get rid of the minus at the first coefficient of the quadratic equation, changing the signs of each term of the equation, which is achieved by multiplying (or dividing) both parts by - 1. For example, from the quadratic equation - 2 x 2 - 3 x + 7 = 0, you can go to a simplified version of it 2 x 2 + 3 x - 7 = 0.

The relationship between roots and coefficients

The already known formula for the roots of quadratic equations x = - b ± D 2 · a expresses the roots of the equation in terms of its numerical coefficients. Based on this formula, we are able to specify other dependencies between roots and coefficients.

The most famous and applicable are the Vieta theorem formulas:

x 1 + x 2 = - b a and x 2 = c a.

In particular, for the given quadratic equation, the sum of the roots is the second coefficient with the opposite sign, and the product of the roots is equal to the free term. For example, by the form of the quadratic equation 3 x 2 - 7 x + 22 = 0, it is possible to immediately determine that the sum of its roots is 7 3, and the product of the roots is 22 3.

You can also find a number of other relationships between the roots and the coefficients of the quadratic equation. For example, the sum of the squares of the roots of a quadratic equation can be expressed in terms of the coefficients:

x 1 2 + x 2 2 = (x 1 + x 2) 2 - 2 x 1 x 2 = - ba 2 - 2 ca = b 2 a 2 - 2 ca = b 2 - 2 a ca 2.

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V modern society the ability to perform actions with equations containing a squared variable can be useful in many areas of activity and is widely used in practice in scientific and technical developments... This is evidenced by the design of sea and river vessels, airplanes and missiles. With the help of such calculations, the trajectories of movement of a variety of bodies, including space objects, are determined. Examples with the solution of quadratic equations are used not only in economic forecasting, in the design and construction of buildings, but also in the most ordinary everyday circumstances. They may be needed on camping trips, at sports events, in stores when shopping, and in other very common situations.

Let's break the expression into its constituent factors

The degree of an equation is determined by the maximum value of the degree of the variable that the expression contains. If it is equal to 2, then such an equation is called square.

If we use the language of formulas, then these expressions, no matter how they look, can always be reduced to the form when the left side of the expression consists of three terms. Among them: ax 2 (that is, a variable squared with its coefficient), bx (an unknown without a square with its coefficient) and c (a free component, that is, an ordinary number). All this on the right side equals 0. In the case when a similar polynomial is missing one of its constituent terms, with the exception of ax 2, it is called an incomplete quadratic equation. Examples with the solution of such problems, the value of the variables in which is easy to find, should be considered first.

If the expression looks in such a way that there are two terms in the expression on the right side, more precisely ax 2 and bx, it is easiest to find x by placing the variable outside the brackets. Now our equation will look like this: x (ax + b). Further, it becomes obvious that either x = 0, or the problem is reduced to finding a variable from the following expression: ax + b = 0. This is dictated by one of the properties of multiplication. The rule is that the product of two factors results in 0 only if one of them is equal to zero.

Example

x = 0 or 8x - 3 = 0

As a result, we get two roots of the equation: 0 and 0.375.

Equations of this kind can describe the movement of bodies under the action of gravity, which began to move from a certain point taken as the origin. Here the mathematical notation takes the following form: y = v 0 t + gt 2/2. Substituting the necessary values, equating the right side to 0 and finding possible unknowns, you can find out the time elapsing from the moment the body rises to the moment it falls, as well as many other quantities. But we'll talk about this later.

Factoring an Expression

The rule described above makes it possible to solve these problems in more complex cases. Let's consider examples with the solution of quadratic equations of this type.

X 2 - 33x + 200 = 0

This square trinomial is complete. First, let's transform the expression and factor it. There are two of them: (x-8) and (x-25) = 0. As a result, we have two roots 8 and 25.

Examples with the solution of quadratic equations in grade 9 allow this method to find a variable in expressions not only of the second, but even the third and fourth orders.

For example: 2x 3 + 2x 2 - 18x - 18 = 0. When factoring the right side into factors with a variable, there are three of them, that is, (x + 1), (x-3) and (x + 3).

As a result, it becomes obvious that this equation has three roots: -3; -1; 3.

Extraction of the square root

Another case of an incomplete second-order equation is an expression represented in the language of letters in such a way that the right-hand side is constructed from the components ax 2 and c. Here, to get the value of the variable, the free term is transferred to right side, and then the square root is extracted from both sides of the equality. It should be noted that in this case, there are usually two roots of the equation. The only exceptions are equalities that do not contain the term c at all, where the variable is equal to zero, as well as variants of expressions when the right-hand side turns out to be negative. In the latter case, there are no solutions at all, since the above actions cannot be performed with roots. Examples of solutions to quadratic equations of this type should be considered.

In this case, the roots of the equation will be the numbers -4 and 4.

Calculation of the area of ​​the land

The need for this kind of calculations appeared in ancient times, because the development of mathematics in many respects in those distant times was due to the need to determine with the greatest accuracy the areas and perimeters of land plots.

Examples with the solution of quadratic equations, compiled on the basis of problems of this kind, should be considered by us.

So, let's say there is a rectangular piece of land, the length of which is 16 meters longer than the width. You should find the length, width and perimeter of the site if you know that its area is 612 m 2.

Getting down to business, let's first draw up the necessary equation. Let's denote by x the width of the section, then its length will be (x + 16). It follows from what has been written that the area is determined by the expression x (x + 16), which, according to the condition of our problem, is 612. This means that x (x + 16) = 612.

The solution of complete quadratic equations, and this expression is just that, cannot be done in the same way. Why? Although the left side of it still contains two factors, their product is not at all equal to 0, so other methods apply here.

Discriminant

First of all, we will make the necessary transformations, then the appearance of this expression will look like this: x 2 + 16x - 612 = 0. This means that we got an expression in the form corresponding to the previously specified standard, where a = 1, b = 16, c = -612.

This can be an example of solving quadratic equations through the discriminant. Here necessary calculations produced according to the scheme: D = b 2 - 4ac. This auxiliary quantity not only makes it possible to find the required quantities in the second-order equation, it determines the number of possible options. If D> 0, there are two of them; for D = 0 there is one root. If D<0, никаких шансов для решения у уравнения вообще не имеется.

About roots and their formula

In our case, the discriminant is: 256 - 4 (-612) = 2704. This indicates that our problem has an answer. If you know, k, the solution of quadratic equations must be continued using the formula below. It allows you to calculate the roots.

This means that in the presented case: x 1 = 18, x 2 = -34. The second option in this dilemma cannot be a solution, because the dimensions of the land plot cannot be measured in negative values, so x (that is, the width of the plot) is 18 m.From here we calculate the length: 18 + 16 = 34, and the perimeter 2 (34+ 18) = 104 (m 2).

Examples and tasks

We continue to study quadratic equations. Examples and a detailed solution to several of them will be given below.

1) 15x 2 + 20x + 5 = 12x 2 + 27x + 1

We transfer everything to the left side of the equality, make a transformation, that is, we get the form of the equation, which is usually called standard, and equate it to zero.

15x 2 + 20x + 5 - 12x 2 - 27x - 1 = 0

Adding similar ones, we define the discriminant: D = 49 - 48 = 1. So our equation will have two roots. We calculate them according to the above formula, which means that the first of them will be 4/3, and the second 1.

2) Now we will reveal the riddles of a different kind.

Let's find out if there are any roots here at all x 2 - 4x + 5 = 1? To obtain an exhaustive answer, let us bring the polynomial to the corresponding familiar form and calculate the discriminant. In this example, the solution of the quadratic equation is not necessary, because the essence of the problem is not at all in this. In this case, D = 16 - 20 = -4, which means that there really are no roots.

Vieta's theorem

It is convenient to solve quadratic equations using the above formulas and the discriminant, when the square root is extracted from the value of the latter. But this is not always the case. However, there are many ways to get the values ​​of variables in this case. Example: solving quadratic equations by Vieta's theorem. She is named after someone who lived in 16th century France and made a brilliant career thanks to his mathematical talent and connections at court. His portrait can be seen in the article.

The pattern noticed by the famous Frenchman was as follows. He proved that the roots of the equation in the sum are numerically equal to -p = b / a, and their product corresponds to q = c / a.

Now let's look at specific tasks.

3x 2 + 21x - 54 = 0

For simplicity, we transform the expression:

x 2 + 7x - 18 = 0

We will use Vieta's theorem, this will give us the following: the sum of the roots is -7, and their product is -18. From this we get that the roots of the equation are the numbers -9 and 2. Having made a check, we will make sure that these values ​​of the variables really fit into the expression.

Parabola graph and equation

The concepts of a quadratic function and quadratic equations are closely related. Examples of this have already been given earlier. Now let's look at some of the math riddles in a little more detail. Any equation of the described type can be visualized. Such a relationship, drawn in the form of a graph, is called a parabola. Its various types are shown in the figure below.

Any parabola has a vertex, that is, a point from which its branches emerge. If a> 0, they go high to infinity, and when a<0, они рисуются вниз. Простейшим примером подобной зависимости является функция y = x 2 . В данном случае в уравнении x 2 =0 неизвестное может принимать только одно значение, то есть х=0, а значит существует только один корень. Это неудивительно, ведь здесь D=0, потому что a=1, b=0, c=0. Выходит формула корней (точнее одного корня) квадратного уравнения запишется так: x = -b/2a.

Visual representations of functions help to solve any equations, including quadratic ones. This method is called graphical. And the value of the variable x is the abscissa coordinate at the points where the graph line intersects with 0x. The coordinates of the vertex can be found by the just given formula x 0 = -b / 2a. And, substituting the obtained value into the original equation of the function, you can find out y 0, that is, the second coordinate of the vertex of the parabola, belonging to the ordinate axis.

The intersection of the branches of the parabola with the abscissa axis

There are a lot of examples with the solution of quadratic equations, but there are also general patterns. Let's consider them. It is clear that the intersection of the graph with the 0x axis for a> 0 is possible only if y 0 takes negative values. And for a<0 координата у 0 должна быть положительна. Для указанных вариантов D>0. Otherwise, D<0. А когда D=0, вершина параболы расположена непосредственно на оси 0х.

The roots can also be determined from the parabola graph. The converse is also true. That is, if it is not easy to get a visual image of a quadratic function, you can equate the right side of the expression to 0 and solve the resulting equation. And knowing the points of intersection with the 0x axis, it is easier to build a graph.

From the history

With the help of equations containing a variable squared, in the old days they did not only do mathematical calculations and determine the areas of geometric shapes. Such calculations were needed by the ancients for grandiose discoveries in the field of physics and astronomy, as well as for making astrological forecasts.

As modern scientists assume, the inhabitants of Babylon were among the first to solve quadratic equations. It happened four centuries before our era. Of course, their calculations were fundamentally different from those currently accepted and turned out to be much more primitive. For example, the Mesopotamian mathematicians had no idea about the existence of negative numbers. They were also unfamiliar with other subtleties that any schoolchild of our time knows.

Perhaps even earlier than the scientists of Babylon, the sage from India Baudhayama took up the solution of quadratic equations. It happened about eight centuries before the advent of the era of Christ. True, the equations of the second order, the methods of solving which he gave, were the simplest. In addition to him, Chinese mathematicians were also interested in similar questions in the old days. In Europe, quadratic equations began to be solved only at the beginning of the 13th century, but later they were used in their works by such great scientists as Newton, Descartes and many others.

», That is, equations of the first degree. In this lesson we will analyze what is called a quadratic equation and how to solve it.

What is called a quadratic equation

Important!

The degree of the equation is determined by the largest degree in which the unknown stands.

If the maximum degree in which the unknown stands is "2", then you have a quadratic equation in front of you.

Examples of quadratic equations

• 5x 2 - 14x + 17 = 0
• −x 2 + x +

  1

  3

  = 0
• x 2 + 0.25x = 0
• x 2 - 8 = 0

Important! The general view of the quadratic equation looks like this:

A x 2 + b x + c = 0

"A", "b" and "c" are given numbers.

• "A" - the first or most significant coefficient;
• "B" is the second coefficient;
• "C" is a free member.

To find "a", "b" and "c" you need to compare your equation with the general form of the quadratic equation "ax 2 + bx + c = 0".

Let's practice defining the coefficients "a", "b" and "c" in quadratic equations.

5x 2 - 14x + 17 = 0 −7x 2 - 13x + 8 = 0 −x 2 + x +

The equation	Odds
a = 5 b = −14 c = 17
a = −7 b = −13 c = 8

1

3

= 0

• a = −1
• b = 1
• c =

  1

  3

x 2 + 0.25x = 0

• a = 1
• b = 0.25
• c = 0

x 2 - 8 = 0

• a = 1
• b = 0
• c = −8

How to solve quadratic equations

Unlike linear equations, to solve quadratic equations, a special formula for finding roots.

Remember!

To solve a quadratic equation you need:

• bring the quadratic equation to the general form "ax 2 + bx + c = 0". That is, only "0" should remain on the right side;
• use formula for roots:

Let's take an example of how to use a formula to find the roots of a quadratic equation. Let's solve the quadratic equation.

X 2 - 3x - 4 = 0

The equation "x 2 - 3x - 4 = 0" has already been reduced to the general form "ax 2 + bx + c = 0" and does not require additional simplifications. To solve it, we just need to apply the formula for finding the roots of a quadratic equation.

Let's define the coefficients "a", "b" and "c" for this equation.

x 1; 2 =
x 1; 2 =
x 1; 2 =
x 1; 2 =

With its help, any quadratic equation is solved.

In the formula "x 1; 2 =" the radical expression is often replaced
"B 2 - 4ac" with the letter "D" and is called the discriminant. The concept of a discriminant is discussed in more detail in the lesson "What is a discriminant".

Consider another example of a quadratic equation.

x 2 + 9 + x = 7x

It is rather difficult to determine the coefficients "a", "b" and "c" in this form. Let's first bring the equation to the general form "ax 2 + bx + c = 0".

X 2 + 9 + x = 7x
x 2 + 9 + x - 7x = 0
x 2 + 9 - 6x = 0
x 2 - 6x + 9 = 0

Now you can use the root formula.

X 1; 2 =
x 1; 2 =
x 1; 2 =
x 1; 2 =
x =

6

2

x = 3
Answer: x = 3

There are times when there are no roots in quadratic equations. This situation occurs when a negative number is found under the root in the formula.

Formulas for the roots of a quadratic equation. The cases of real, multiple and complex roots are considered. Factoring a square trinomial. Geometric interpretation. Examples of determining roots and factoring.

Content

See also: Solving quadratic equations online

Basic formulas

Consider a quadratic equation:
(1) .
Quadratic Roots(1) are determined by the formulas:
; .
These formulas can be combined like this:
.
When the roots of the quadratic equation are known, then the second degree polynomial can be represented as a product of factors (factorized):
.

Further, we assume that are real numbers.
Consider quadratic discriminant:
.
If the discriminant is positive, then the quadratic equation (1) has two different real roots:
; .
Then the factorization of the square trinomial is:
.
If the discriminant is zero, then the quadratic equation (1) has two multiple (equal) real roots:
.
Factorization:
.
If the discriminant is negative, then the quadratic equation (1) has two complex conjugate roots:
;
.
Here is an imaginary unit,;
and - real and imaginary parts of the roots:
; .
Then

.

Graphic interpretation

If you plot the function
,
which is a parabola, then the points of intersection of the graph with the axis will be the roots of the equation
.
When, the graph crosses the abscissa axis (axis) at two points ().
When, the graph touches the abscissa axis at one point ().
When, the graph does not cross the abscissa axis ().

Useful Quadratic Equations

(f.1) ;
(f.2) ;
(f.3) .

Derivation of the formula for the roots of a quadratic equation

We perform transformations and apply formulas (f.1) and (f.3):




,
where
; .

So, we got the formula for the second degree polynomial in the form:
.
Hence it is seen that the equation

performed at
and .
That is, they are the roots of the quadratic equation
.

Examples of determining the roots of a quadratic equation

Example 1

(1.1) .

.
Comparing with our equation (1.1), we find the values ​​of the coefficients:
.
We find the discriminant:
.
Since the discriminant is positive, the equation has two real roots:
;
;
.

From this we get the factorization of the square trinomial:

.

Function graph y = 2 x 2 + 7 x + 3 crosses the abscissa axis at two points.

Let's plot the function
.
The graph of this function is a parabola. It crosses the abscissa axis (axis) at two points:
and .
These points are the roots of the original equation (1.1).

;
;
.

Example 2

Find the roots of a quadratic equation:
(2.1) .

Let's write the quadratic equation in general form:
.
Comparing with the original equation (2.1), we find the values ​​of the coefficients:
.
We find the discriminant:
.
Since the discriminant is zero, the equation has two multiple (equal) roots:
;
.

Then the factorization of the trinomial is:
.

Function graph y = x 2 - 4 x + 4 touches the abscissa axis at one point.

Let's plot the function
.
The graph of this function is a parabola. It touches the abscissa axis (axis) at one point:
.
This point is the root of the original equation (2.1). Since this root enters the factorization two times:
,
then such a root is usually called multiple. That is, they believe that there are two equal roots:
.

;
.

Example 3

Find the roots of a quadratic equation:
(3.1) .

Let's write the quadratic equation in general form:
(1) .
Let us rewrite the original equation (3.1):
.
Comparing with (1), we find the values ​​of the coefficients:
.
We find the discriminant:
.
The discriminant is negative,. Therefore, there are no valid roots.

Complex roots can be found:
;
;
.

Then


.

The graph of the function does not cross the abscissa axis. There are no valid roots.

Let's plot the function
.
The graph of this function is a parabola. It does not cross the abscissa axis (axis). Therefore, there are no valid roots.

There are no valid roots. Complex roots:
;
;
.

See also: