We continue to study the topic solution of equations". We have already got acquainted with linear equations and now we are going to get acquainted with quadratic equations.

First, we will discuss 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. Next, let's move on to solving the complete equations, get the formula for the roots, get acquainted with the discriminant of the quadratic equation and consider the solutions characteristic examples. Finally, we trace the connections 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 definitions related to it. After that, you can consider the main types of quadratic equations: reduced and non-reduced, 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 different from zero.

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 us 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.

Numbers a , b and c are called coefficients of the quadratic equation a x 2 + b x + c \u003d 0, and the coefficient a is called the first, or senior, or coefficient at x 2, b is the second coefficient, or coefficient at x, and c is a free member.

For example, let's take a quadratic equation of the form 5 x 2 −2 x−3=0, here the leading coefficient is 5, the second coefficient is −2, and the free term is −3. Note that when the coefficients b and/or c are negative, as in the example just given, then short form writing a quadratic equation of the form 5 x 2 −2 x−3=0 , and 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 notation of the quadratic equation, which is due to the peculiarities of the notation of such . For example, in the quadratic equation y 2 −y+3=0, the leading coefficient is one, and the coefficient at y is −1.

Reduced and non-reduced quadratic equations

Depending on the value of the leading coefficient, reduced and non-reduced quadratic equations are distinguished. 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. - reduced, 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 of its parts 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 non-reduced quadratic equation, or, like it, has no roots.

Let's take an example of 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 perform the division of both parts of the original equation by the leading coefficient 3, it is non-zero, so we can perform this action. We have (3 x 2 +12 x−7):3=0:3 , which is the same as (3 x 2):3+(12 x):3−7:3=0 , and so on (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

There is a condition a≠0 in the definition of a quadratic equation. This condition is necessary in order for the equation a x 2 +b x+c=0 to be exactly square, since with 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 equal to 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 its turn

Definition.

Complete quadratic equation is an equation in which all coefficients are different from zero.

These names are not given by chance. This will become clear from the following discussion.

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 the 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

It follows from the information of the previous paragraph that there is three kinds of incomplete quadratic equations:

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

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

a x 2 \u003d 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 its both parts by a non-zero number a. Obviously, the root of the equation x 2 \u003d 0 is zero, since 0 2 \u003d 0. This equation has no other roots, which is explained, indeed, for any non-zero number p, the inequality p 2 >0 takes place, which implies that for p≠0, the equality p 2 =0 is never achieved.

So, the incomplete quadratic equation a x 2 \u003d 0 has a single root x \u003d 0.

As an example, we give the solution of an incomplete quadratic equation −4·x 2 =0. It is equivalent to the equation x 2 \u003d 0, its only root is x \u003d 0, therefore, the original equation has a single root zero.

A short solution in this case can be issued in the following way:
−4 x 2 \u003d 0,
x 2 \u003d 0,
x=0 .

a x 2 +c=0

Now consider how incomplete quadratic equations are solved, in which the coefficient b is equal to zero, and c≠0, that is, equations of the form a x 2 +c=0. We know that the transfer of a term from one side of the equation to the other with the opposite sign, as well as the division of both sides of the equation by a non-zero number, give an equivalent equation. Therefore, the following equivalent transformations of the incomplete quadratic equation a x 2 +c=0 can be carried out:

• move c to the right side, which gives the equation a x 2 =−c,
• and divide both 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 , because by condition c≠0 . We will separately analyze 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 we recall about, then the root of the equation immediately becomes obvious, it is the 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 contradiction. Let's do it.

Let's denote the just voiced roots of the equation as x 1 and −x 1 . Suppose that the equation has another root x 2 different from the indicated roots x 1 and −x 1 . It is known that substitution into the equation instead of x of its roots 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 operations with numbers allow us to rewrite the resulting equality as (x 1 − x 2)·(x 1 + x 2)=0 . We know that the product of two numbers is equal to zero if and only if at least one of them is equal to 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 . So we have come 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 other roots than and .

Let's summarize the information in this paragraph. The incomplete quadratic equation a x 2 +c=0 is equivalent to the equation , which

• 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 a negative number is obtained on the right side, this equation has no roots, therefore, the original incomplete quadratic equation 9 x 2 +7=0 has no roots.

Let's solve one more incomplete quadratic equation −x 2 +9=0. We transfer the nine to the right side: -x 2 \u003d -9. Now we divide both parts by −1, we get x 2 =9. The right side contains a positive number, from which we conclude that or . After 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 take the common factor x out of brackets. This allows us to move 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 set 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.

We take x out of brackets, this gives the equation. It is equivalent to two equations x=0 and . We solve the resulting linear equation: , and after dividing the mixed number by an ordinary fraction, we find . Therefore, the roots of the original equation are x=0 and .

After getting the necessary practice, the solutions of such equations can be written briefly:

Answer:

x=0 , .

Discriminant, formula of the roots of a quadratic equation

To solve quadratic equations, there is a root formula. Let's write down the formula of the roots of the quadratic equation: , where D=b 2 −4 a c- so-called discriminant of a quadratic equation. The notation essentially means that .

It is useful to know how the root formula was obtained, and how it is applied in finding the roots of quadratic equations. Let's deal with this.

Derivation of the formula of the roots of a quadratic equation

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

• We can divide both parts of this equation by a non-zero number a, as a result we get the reduced quadratic equation.
• Now select a full 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 side with the opposite sign, we have .
• And let's also transform the expression on the right side: .

As a result, we arrive at the equation , which 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 . 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, , from which its only root is visible;
• if , then or , which is the same as 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 is called discriminant of a quadratic equation and marked with the letter D. From here, 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.

We return to the equation , rewrite it using the notation of the discriminant: . And we conclude:

• 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 can be rewritten in the form or , and after expanding and reducing the fractions to a common denominator, we get .

So we derived the formulas for the roots of the quadratic equation, they look like , 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 a quadratic equation. When the discriminant is equal to zero, both formulas give the same root value corresponding to the only solution of 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 from a negative number, which takes us beyond and school curriculum. With a negative discriminant, the quadratic equation has no real roots, but has a pair complex conjugate roots, which can be found using the same root formulas we obtained.

Algorithm for solving quadratic equations using root formulas

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

However, in a school algebra course, it is usually we are talking 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 after that calculate the values ​​of the roots.

The above reasoning allows us to write algorithm for solving a quadratic equation. To solve the quadratic equation a x 2 + b x + c \u003d 0, you need:

• using 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 using the formula if D=0 ;
• find two real roots of a quadratic equation using the root formula if the discriminant is positive.

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

You can move on to examples of applying the algorithm for solving quadratic equations.

Examples of solving quadratic equations

Consider solutions of three quadratic equations with positive, negative, and zero discriminant. 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, you first 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, the quadratic equation has two real roots. Let's find them by the formula of roots , we get , here we can simplify the expressions obtained by doing factoring out the sign of the root followed by fraction reduction:

Answer:

Let's move on to the next typical example.

Example.

Solve the quadratic equation −4 x 2 +28 x−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 specify complex roots, then we use the well-known formula for the roots of the quadratic equation, and perform operations with complex numbers:

Answer:

there are no real roots, the complex roots are: .

Once again, we note that if the discriminant of the quadratic equation is negative, then the school usually immediately writes down the answer, in which they indicate that there are no real roots, and they do not find complex roots.

Root formula for even second coefficients

The formula for the roots of a quadratic equation , where D=b 2 −4 a c allows you to get a more compact formula that allows you to solve quadratic equations with an even coefficient at x (or simply with a coefficient that looks like 2 n, for example, or 14 ln5=2 7 ln5 ). Let's take her 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 known to us. To do this, we 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 root formula:

Denote the expression n 2 − a c as D 1 (sometimes it is denoted 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 D 1 is also an indicator of the presence or absence of the roots of the quadratic equation.

So, to solve a 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 using the formula;
• If D 1 >0, then find two real roots using the formula.

Consider the solution of the example using the root formula obtained in this paragraph.

Example.

Solve the quadratic equation 5 x 2 −6 x−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. We 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:

Simplification of the form of quadratic equations

Sometimes, before embarking on the calculation of the roots of a quadratic equation using 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 sides of it by some number. For example, in the previous paragraph, we managed to achieve a simplification of the equation 1100 x 2 −400 x −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 parts 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. absolute values ​​of its coefficients: gcd(12, 42, 48)= gcd(gcd(12, 42), 48)= gcd(6, 48)=6 . Dividing both parts 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 parts of the quadratic equation is usually done to get rid of fractional coefficients. In this case, the multiplication is carried out on the denominators of its coefficients. For example, if both parts of a quadratic equation are multiplied by LCM(6, 3, 1)=6 , then it will take 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 highest coefficient of the quadratic equation by changing the signs of all terms, which corresponds to multiplying (or dividing) both parts by −1. For example, usually from the quadratic equation −2·x 2 −3·x+7=0 go to the solution 2·x 2 +3·x−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 formula of the roots, you can get other relationships between the roots and coefficients.

The most well-known and applicable formulas from the Vieta 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 the free term. For example, by the form of the quadratic equation 3 x 2 −7 x+22=0, you 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 coefficients of the quadratic equation. For example, you can express the sum of the squares of the roots of a quadratic equation in terms of its coefficients: .

Bibliography.

• Algebra: textbook for 8 cells. 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.
• Mordkovich A. G. Algebra. 8th grade. At 2 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemozina, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.

The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Equations have been used by man since ancient times and since then their use has only increased. The discriminant allows you to solve any quadratic equations using the general formula, which has the following form:

The discriminant formula depends on the degree of the polynomial. The above formula is suitable for solving quadratic equations of the following form:

The discriminant has the following properties that you need to know:

* "D" is 0 when the polynomial has multiple roots (equal roots);

* "D" is a symmetric polynomial with respect to the roots of the polynomial and therefore is a polynomial in its coefficients; moreover, the coefficients of this polynomial are integers, regardless of the extension in which the roots are taken.

Suppose we are given a quadratic equation of the following form:

1 equation

According to the formula we have:

Since \, then the equation has 2 roots. Let's define them:

Where can I solve the equation through the discriminant online solver?

You can solve the equation on our website https: // site. Free online solver will allow you to solve an online equation of any complexity in seconds. All you have to do is just enter your data into the solver. You can also watch the video instruction and learn how to solve the equation on our website. And if you have any questions, you can ask them in our Vkontakte group http://vk.com/pocketteacher. Join our group, we are always happy to help you.

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AT modern society the ability to operate 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 can be evidenced by the design of sea and river vessels, aircraft and missiles. With the help of such calculations, the trajectories of movement of the most different bodies, including space objects. 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 component factors

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

If we speak in the language of formulas, then these expressions, no matter how they look, can always be brought 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 (free component, that is, an ordinary number). All this is equal to 0 on the right side. In the case when such a polynomial does not have 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, in which the value of the variables is not difficult to find, should be considered first.

If the expression looks like it has two terms on the right side of the expression, more precisely ax 2 and bx, it is easiest to find x by bracketing the variable. 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 says that the product of two factors results in 0 only if one of them is 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. By substituting the necessary values, equating the right side to 0 and finding possible unknowns, you can find out the time elapsed from the moment the body rises to the moment it falls, as well as many other quantities. But we will talk about this later.

Factoring an Expression

The rule described above makes it possible to solve these problems and in more difficult cases. Consider examples with the solution of quadratic equations of this type.

X2 - 33x + 200 = 0

This square trinomial is complete. First, we transform the expression and decompose it into factors. 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 of 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; -one; 3.

Extracting the square root

Another case of an incomplete second-order equation is an expression written in the language of letters in such a way that the right side is built from the components ax 2 and c. Here, to get the value of the variable, the free term is transferred to right side, and after that, from both parts of the equality, Square root. It should be noted that in this case There are usually two roots of an 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 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 ​​land

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

We should also consider examples with the solution of quadratic equations compiled on the basis of problems of this kind.

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

Getting down to business, at first we will make the necessary equation. Let's denote the width of the section as x, 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) \u003d 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, the product of them is not equal to 0 at all, so other methods are used here.

Discriminant

First of all, we make the necessary transformations, then appearance this expression will look like this: x 2 + 16x - 612 = 0. This means that we have received 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 value not only makes it possible to find the desired values ​​in the second-order equation, it determines the number of possible options. In case D>0, there are two of them; for D=0 there is one root. In case 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, to, 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 size of the land plot cannot be measured in negative values, which means that 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 the study of quadratic equations. Examples and a detailed solution of several of them will be given below.

1) 15x2 + 20x + 5 = 12x2 + 27x + 1

Let's 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 the standard one, and equate it to zero.

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

Having added similar ones, we determine the discriminant: D \u003d 49 - 48 \u003d 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 equal to 4/3, and the second 1.

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

Let's find out if there are roots x 2 - 4x + 5 = 1 here at all? To obtain an exhaustive answer, we bring the polynomial to the corresponding familiar form and calculate the discriminant. In this example, it is not necessary to solve the quadratic equation, because the essence of the problem is not at all in this. In this case, D \u003d 16 - 20 \u003d -4, which means that there really are no roots.

Vieta's theorem

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

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

Now let's look at specific tasks.

3x2 + 21x - 54 = 0

For simplicity, let's transform the expression:

x 2 + 7x - 18 = 0

Using the Vieta theorem, this will give us the following: the sum of the roots is -7, and their product is -18. From here 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.

Graph and Equation of a Parabola

The concepts of a quadratic function and quadratic equations are closely related. Examples of this have already been given previously. Now let's look at some mathematical puzzles in a little more detail. Any equation of the described type can be represented visually. Such a dependence, 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 come out. 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 graphic. And the value of the x variable is the abscissa coordinate at the points where the graph line intersects with 0x. The coordinates of the vertex can be found by the formula just given x 0 = -b / 2a. And, substituting the resulting value into the original equation of the function, you can find out y 0, that is, the second coordinate of the parabola vertex belonging to the y-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.

From the graph of a parabola, you can also determine the roots. The reverse is also true. That is, if it is not easy to get a visual representation 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 plot.

From the history

With the help of equations containing a squared variable, in the old days, not only did mathematical calculations and determined the area of ​​\u200b\u200bgeometric shapes. The ancients needed such calculations for grandiose discoveries in the field of physics and astronomy, as well as for making astrological forecasts.

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

Perhaps even earlier than the scientists of Babylon, the sage from India, Baudhayama, took up the solution of quadratic equations. This happened about eight centuries before the advent of the era of Christ. True, the second-order equations, the methods for 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 work by such great scientists as Newton, Descartes and many others.