Problems on the quadratic equation are also studied in school curriculum and in universities. They are understood as equations of the form a * x ^ 2 + b * x + c \u003d 0, where x- variable, a,b,c – constants; a<>0 . The problem is to find the roots of the equation.

The geometric meaning of the quadratic equation

The graph of a function that is represented by a quadratic equation is a parabola. The solutions (roots) of a quadratic equation are the points of intersection of the parabola with the x-axis. It follows that there are three possible cases:
1) the parabola has no points of intersection with the x-axis. This means that it is in the upper plane with branches up or the lower one with branches down. In such cases, the quadratic equation has no real roots (it has two complex roots).

2) the parabola has one point of intersection with the axis Ox. Such a point is called the vertex of the parabola, and the quadratic equation in it acquires its minimum or maximum value. In this case, the quadratic equation has one real root (or two identical roots).

3) The last case is more interesting in practice - there are two points of intersection of the parabola with the abscissa axis. This means that there are two real roots of the equation.

Based on the analysis of the coefficients at the powers of the variables, interesting conclusions can be drawn about the placement of the parabola.

1) If the coefficient a is greater than zero, then the parabola is directed upwards, if negative, the branches of the parabola are directed downwards.

2) If the coefficient b is greater than zero, then the vertex of the parabola lies in the left half-plane, if it takes a negative value, then in the right.

Derivation of a formula for solving a quadratic equation

Let's transfer the constant from the quadratic equation

for the equal sign, we get the expression

Multiply both sides by 4a

To get a full square on the left, add b ^ 2 in both parts and perform the transformation

From here we find

Formula of the discriminant and roots of the quadratic equation

The discriminant is the value of the radical expression. If it is positive, then the equation has two real roots, calculated by the formula When the discriminant is zero, the quadratic equation has one solution (two coinciding roots), which are easy to obtain from the above formula for D=0. When the discriminant is negative, there are no real roots. However, to study the solutions of the quadratic equation in the complex plane, and their value is calculated by the formula

Vieta's theorem

Consider two roots of a quadratic equation and construct a quadratic equation on their basis. The Vieta theorem itself easily follows from the notation: if we have a quadratic equation of the form then the sum of its roots is equal to the coefficient p, taken with the opposite sign, and the product of the roots of the equation is equal to the free term q. The formula for the above will look like If the constant a in the classical equation is nonzero, then you need to divide the entire equation by it, and then apply the Vieta theorem.

Schedule of the quadratic equation on factors

Let the task be set: to decompose the quadratic equation into factors. To perform it, we first solve the equation (find the roots). Next, we substitute the found roots into the formula for expanding the quadratic equation. This problem will be solved.

Tasks for a quadratic equation

Task 1. Find the roots of a quadratic equation

x^2-26x+120=0 .

Solution: Write down the coefficients and substitute in the discriminant formula

root of given value equal to 14, it is easy to find it with a calculator, or remember it with frequent use, however, for convenience, at the end of the article I will give you a list of squares of numbers that can often be found in such tasks.
The found value is substituted into the root formula

and we get

Task 2. solve the equation

2x2+x-3=0.

Solution: We have a complete quadratic equation, write out the coefficients and find the discriminant


By known formulas find the roots of the quadratic equation

Task 3. solve the equation

9x2 -12x+4=0.

Solution: We have a complete quadratic equation. Determine the discriminant

We got the case when the roots coincide. We find the values ​​​​of the roots by the formula

Task 4. solve the equation

x^2+x-6=0 .

Solution: In cases where there are small coefficients for x, it is advisable to apply the Vieta theorem. By its condition, we obtain two equations

From the second condition, we get that the product must be equal to -6. This means that one of the roots is negative. We have the following possible pair of solutions(-3;2), (3;-2) . Taking into account the first condition, we reject the second pair of solutions.
The roots of the equation are

Task 5. Find the lengths of the sides of a rectangle if its perimeter is 18 cm and area is 77 cm 2.

Solution: Half the perimeter of a rectangle is equal to the sum of the adjacent sides. Let's denote x - big side, then 18-x is its smaller side. The area of ​​a rectangle is equal to the product of these lengths:
x(18x)=77;
or
x 2 -18x + 77 \u003d 0.
Find the discriminant of the equation

We calculate the roots of the equation

If a x=11, then 18x=7 , vice versa is also true (if x=7, then 21-x=9).

Problem 6. Factorize the quadratic 10x 2 -11x+3=0 equation.

Solution: Calculate the roots of the equation, for this we find the discriminant

We substitute the found value into the formula of the roots and calculate

We apply the formula for expanding the quadratic equation in terms of roots

Expanding the brackets, we get the identity.

Quadratic equation with parameter

Example 1. For what values ​​of the parameter a , does the equation (a-3) x 2 + (3-a) x-1 / 4 \u003d 0 have one root?

Solution: By direct substitution of the value a=3, we see that it has no solution. Further, we will use the fact that with a zero discriminant, the equation has one root of multiplicity 2. Let's write out the discriminant

simplify it and equate to zero

We have obtained a quadratic equation with respect to the parameter a, the solution of which is easy to obtain using the Vieta theorem. The sum of the roots is 7, and their product is 12. By simple enumeration, we establish that the numbers 3.4 will be the roots of the equation. Since we have already rejected the solution a=3 at the beginning of the calculations, the only correct one will be - a=4. Thus, for a = 4, the equation has one root.

Example 2. For what values ​​of the parameter a , the equation a(a+3)x^2+(2a+6)x-3a-9=0 has more than one root?

Solution: Consider first the singular points, they will be the values ​​a=0 and a=-3. When a=0, the equation will be simplified to the form 6x-9=0; x=3/2 and there will be one root. For a= -3 we get the identity 0=0 .
Calculate the discriminant

and find the values ​​of a for which it is positive

From the first condition we get a>3. For the second, we find the discriminant and the roots of the equation


Let's define the intervals where the function takes positive values. By substituting the point a=0 we get 3>0 . So, outside the interval (-3; 1/3) the function is negative. Don't forget the dot a=0 which should be excluded, since the original equation has one root in it.
As a result, we obtain two intervals that satisfy the condition of the problem

There will be many similar tasks in practice, try to deal with the tasks yourself and do not forget to take into account the conditions that are mutually exclusive. Study well the formulas for solving quadratic equations, they are quite often needed in calculations in various problems and sciences.

AT modern society the ability to perform operations 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 the movement of various 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 in hiking trips, at sports, 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 on the right side is equal to 0. 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 in such a way that there are 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 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 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 given 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 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 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 is largely in those distant times was 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 rectangular area 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 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 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 quadratic function and quadratic equations closely connected. 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.

Quadratic equations are studied in grade 8, so there is nothing complicated here. The ability to solve them is essential.

A quadratic equation is an equation of the form ax 2 + bx + c = 0, where the coefficients a , b and c are arbitrary numbers, and a ≠ 0.

Before studying specific solution methods, we note that all quadratic equations can be divided into three classes:

1. Have no roots;
2. They have exactly one root;
3. They have two different roots.

This is an important difference between quadratic and linear equations, where the root always exists and is unique. How to determine how many roots an equation has? There is a wonderful thing for this - discriminant.

Discriminant

Let the quadratic equation ax 2 + bx + c = 0 be given. Then the discriminant is simply the number D = b 2 − 4ac .

This formula must be known by heart. Where it comes from is not important now. Another thing is important: by the sign of the discriminant, you can determine how many roots a quadratic equation has. Namely:

1. If D< 0, корней нет;
2. If D = 0, there is exactly one root;
3. If D > 0, there will be two roots.

Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many people think. Take a look at the examples and you will understand everything yourself:

Task. How many roots do quadratic equations have:

1. x 2 - 8x + 12 = 0;
2. 5x2 + 3x + 7 = 0;
3. x 2 − 6x + 9 = 0.

We write the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 − 4 1 12 = 64 − 48 = 16

So, the discriminant is positive, so the equation has two different roots. We analyze the second equation in the same way:
a = 5; b = 3; c = 7;
D \u003d 3 2 - 4 5 7 \u003d 9 - 140 \u003d -131.

The discriminant is negative, there are no roots. The last equation remains:
a = 1; b = -6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.

The discriminant is equal to zero - the root will be one.

Note that coefficients have been written out for each equation. Yes, it's long, yes, it's tedious - but you won't mix up the odds and don't make stupid mistakes. Choose for yourself: speed or quality.

By the way, if you “fill your hand”, after a while you will no longer need to write out all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not so many.

The roots of a quadratic equation

Now let's move on to the solution. If the discriminant D > 0, the roots can be found using the formulas:

The basic formula for the roots of a quadratic equation

When D = 0, you can use any of these formulas - you get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.

1. x 2 - 2x - 3 = 0;
2. 15 - 2x - x2 = 0;
3. x2 + 12x + 36 = 0.

First equation:
x 2 - 2x - 3 = 0 ⇒ a = 1; b = −2; c = -3;
D = (−2) 2 − 4 1 (−3) = 16.

D > 0 ⇒ the equation has two roots. Let's find them:

Second equation:
15 − 2x − x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 − 4 (−1) 15 = 64.

D > 0 ⇒ the equation again has two roots. Let's find them

\[\begin(align) & ((x)_(1))=\frac(2+\sqrt(64))(2\cdot \left(-1 \right))=-5; \\ & ((x)_(2))=\frac(2-\sqrt(64))(2\cdot \left(-1 \right))=3. \\ \end(align)\]

Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 − 4 1 36 = 0.

D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:

As you can see from the examples, everything is very simple. If you know the formulas and be able to count, there will be no problems. Most often, errors occur when negative coefficients are substituted into the formula. Here, again, the technique described above will help: look at the formula literally, paint each step - and get rid of mistakes very soon.

Incomplete quadratic equations

It happens that the quadratic equation is somewhat different from what is given in the definition. For example:

1. x2 + 9x = 0;
2. x2 − 16 = 0.

It is easy to see that one of the terms is missing in these equations. Such quadratic equations are even easier to solve than standard ones: they do not even need to calculate the discriminant. So let's introduce a new concept:

The equation ax 2 + bx + c = 0 is called an incomplete quadratic equation if b = 0 or c = 0, i.e. the coefficient of the variable x or the free element is equal to zero.

Of course, a very difficult case is possible when both of these coefficients are equal to zero: b \u003d c \u003d 0. In this case, the equation takes the form ax 2 \u003d 0. Obviously, such an equation has a single root: x \u003d 0.

Let's consider other cases. Let b \u003d 0, then we get an incomplete quadratic equation of the form ax 2 + c \u003d 0. Let's slightly transform it:

Since the arithmetic square root exists only from a non-negative number, the last equality only makes sense when (−c / a ) ≥ 0. Conclusion:

1. If an incomplete quadratic equation of the form ax 2 + c = 0 satisfies the inequality (−c / a ) ≥ 0, there will be two roots. The formula is given above;
2. If (−c / a )< 0, корней нет.

As you can see, the discriminant was not required - there are no complex calculations at all in incomplete quadratic equations. In fact, it is not even necessary to remember the inequality (−c / a ) ≥ 0. It is enough to express the value of x 2 and see what is on the other side of the equal sign. If there is a positive number, there will be two roots. If negative, there will be no roots at all.

Now let's deal with equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factorize the polynomial:

Taking the common factor out of the bracket

The product is equal to zero when at least one of the factors is equal to zero. This is where the roots come from. In conclusion, we will analyze several of these equations:

Task. Solve quadratic equations:

1. x2 − 7x = 0;
2. 5x2 + 30 = 0;
3. 4x2 − 9 = 0.

x 2 − 7x = 0 ⇒ x (x − 7) = 0 ⇒ x 1 = 0; x2 = −(−7)/1 = 7.

5x2 + 30 = 0 ⇒ 5x2 = -30 ⇒ x2 = -6. There are no roots, because the square cannot be equal to a negative number.

4x 2 − 9 = 0 ⇒ 4x 2 = 9 ⇒ x 2 = 9/4 ⇒ x 1 = 3/2 = 1.5; x 2 \u003d -1.5.

Quadratic equations. Discriminant. Solution, examples.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

Types of quadratic equations

What is a quadratic equation? What does it look like? In term quadratic equation keyword is "square". It means that in the equation necessarily there must be an x ​​squared. In addition to it, in the equation there may be (or may not be!) Just x (to the first degree) and just a number (free member). And there should not be x's in a degree greater than two.

In mathematical terms, a quadratic equation is an equation of the form:

Here a, b and c- some numbers. b and c- absolutely any, but a- anything but zero. For example:

Here a =1; b = 3; c = -4

Here a =2; b = -0,5; c = 2,2

Here a =-3; b = 6; c = -18

Well, you get the idea...

In these quadratic equations, on the left, there is full set members. x squared with coefficient a, x to the first power with coefficient b and free member of

Such quadratic equations are called complete.

And if b= 0, what will we get? We have X will disappear in the first degree. This happens from multiplying by zero.) It turns out, for example:

5x 2 -25 = 0,

2x 2 -6x=0,

-x 2 +4x=0

Etc. And if both coefficients b and c are equal to zero, then it is even simpler:

2x 2 \u003d 0,

-0.3x 2 \u003d 0

Such equations, where something is missing, are called incomplete quadratic equations. Which is quite logical.) Please note that x squared is present in all equations.

By the way why a can't be zero? And you substitute instead a zero.) The X in the square will disappear! The equation will become linear. And it's done differently...

That's all the main types of quadratic equations. Complete and incomplete.

Solution of quadratic equations.

Solution of complete quadratic equations.

Quadratic equations are easy to solve. By formulas and clear simple rules. At the first stage, you need given equation lead to standard view, i.e. to the view:

If the equation is already given to you in this form, you do not need to do the first stage.) The main thing is to correctly determine all the coefficients, a, b and c.

The formula for finding the roots of a quadratic equation looks like this:

The expression under the root sign is called discriminant. But more about him below. As you can see, to find x, we use only a, b and c. Those. coefficients from the quadratic equation. Just carefully substitute the values a, b and c into this formula and count. Substitute with your signs! For example, in the equation:

a =1; b = 3; c= -4. Here we write:

Example almost solved:

This is the answer.

Everything is very simple. And what do you think, you can't go wrong? Well, yes, how...

The most common mistakes are confusion with the signs of values a, b and c. Or rather, not with their signs (where is there to get confused?), But with the substitution negative values into the formula for calculating the roots. Here, a detailed record of the formula with specific numbers saves. If there are problems with calculations, so do it!

Suppose we need to solve the following example:

Here a = -6; b = -5; c = -1

Let's say you know that you rarely get answers the first time.

Well, don't be lazy. It will take 30 seconds to write an extra line. And the number of errors will drop sharply. So we write in detail, with all the brackets and signs:

It seems incredibly difficult to paint so carefully. But it only seems. Try it. Well, or choose. Which is better, fast, or right? Besides, I will make you happy. After a while, there will be no need to paint everything so carefully. It will just turn out right. Especially if you apply practical techniques, which are described below. This evil example with a bunch of minuses will be solved easily and without errors!

But, often, quadratic equations look slightly different. For example, like this:

Did you know?) Yes! This is incomplete quadratic equations.

Solution of incomplete quadratic equations.

They can also be solved by the general formula. You just need to correctly figure out what is equal here a, b and c.

Realized? In the first example a = 1; b = -4; a c? It doesn't exist at all! Well, yes, that's right. In mathematics, this means that c = 0 ! That's all. Substitute zero into the formula instead of c, and everything will work out for us. Similarly with the second example. Only zero we don't have here with, a b !

But incomplete quadratic equations can be solved much easier. Without any formulas. Consider the first incomplete equation. What can be done on the left side? You can take the X out of brackets! Let's take it out.

And what from this? And the fact that the product is equal to zero if, and only if any of the factors is equal to zero! Don't believe? Well, then come up with two non-zero numbers that, when multiplied, will give zero!
Does not work? Something...
Therefore, we can confidently write: x 1 = 0, x 2 = 4.

Everything. These will be the roots of our equation. Both fit. When substituting any of them into the original equation, we get the correct identity 0 = 0. As you can see, the solution is much simpler than the general formula. I note, by the way, which X will be the first, and which the second - it is absolutely indifferent. Easy to write in order x 1- whichever is less x 2- that which is more.

The second equation can also be easily solved. We move 9 to the right side. We get:

It remains to extract the root from 9, and that's it. Get:

also two roots . x 1 = -3, x 2 = 3.

This is how all incomplete quadratic equations are solved. Either by taking x out of brackets, or simple transfer numbers to the right, followed by root extraction.
It is extremely difficult to confuse these methods. Simply because in the first case you will have to extract the root from X, which is somehow incomprehensible, and in the second case there is nothing to take out of brackets ...

Discriminant. Discriminant formula.

Magic word discriminant ! A rare high school student has not heard this word! The phrase “decide through the discriminant” is reassuring and reassuring. Because there is no need to wait for tricks from the discriminant! It is simple and trouble-free in handling.) I remind you of the most general formula for solutions any quadratic equations:

The expression under the root sign is called the discriminant. The discriminant is usually denoted by the letter D. Discriminant formula:

D = b 2 - 4ac

And what is so special about this expression? Why does it deserve a special name? What meaning of the discriminant? After all -b, or 2a in this formula they don’t specifically name ... Letters and letters.

The point is this. When solving a quadratic equation using this formula, it is possible only three cases.

1. The discriminant is positive. This means that you can extract the root from it. Whether the root is extracted well or badly is another question. It is important what is extracted in principle. Then your quadratic equation has two roots. Two different solutions.

2. The discriminant is zero. Then you have one solution. Since adding or subtracting zero in the numerator does not change anything. Strictly speaking, this is not a single root, but two identical. But, in a simplified version, it is customary to talk about one solution.

3. The discriminant is negative. A negative number does not take the square root. Well, okay. This means there are no solutions.

To be honest, at simple solution quadratic equations, the concept of discriminant is not particularly required. We substitute the values ​​​​of the coefficients in the formula, and we consider. There everything turns out by itself, and two roots, and one, and not a single one. However, when solving more difficult tasks, without knowledge meaning and discriminant formula not enough. Especially - in equations with parameters. Such equations are aerobatics for the GIA and the Unified State Examination!)

So, how to solve quadratic equations through the discriminant you remembered. Or learned, which is also not bad.) You know how to correctly identify a, b and c. Do you know how attentively substitute them into the root formula and attentively count the result. Did you understand that the key word here is - attentively?

Now take note of the practical techniques that dramatically reduce the number of errors. The very ones that are due to inattention ... For which it is then painful and insulting ...

First reception . Do not be lazy before solving a quadratic equation to bring it to a standard form. What does this mean?
Suppose, after any transformations, you get the following equation:

Do not rush to write the formula of the roots! You will almost certainly mix up the odds a, b and c. Build the example correctly. First, x squared, then without a square, then a free member. Like this:

And again, do not rush! The minus before the x squared can upset you a lot. Forgetting it is easy... Get rid of the minus. How? Yes, as taught in the previous topic! We need to multiply the whole equation by -1. We get:

And now you can safely write down the formula for the roots, calculate the discriminant and complete the example. Decide on your own. You should end up with roots 2 and -1.

Second reception. Check your roots! According to Vieta's theorem. Don't worry, I'll explain everything! Checking last thing the equation. Those. the one by which we wrote down the formula of the roots. If (as in this example) the coefficient a = 1, check the roots easily. It is enough to multiply them. You should get a free term, i.e. in our case -2. Pay attention, not 2, but -2! free member with your sign . If it didn’t work out, it means they already messed up somewhere. Look for an error.

If it worked out, you need to fold the roots. Last and final check. Should be a ratio b with opposite sign. In our case -1+2 = +1. A coefficient b, which is before the x, is equal to -1. So, everything is right!
It is a pity that it is so simple only for examples where x squared is pure, with a coefficient a = 1. But at least check in such equations! Everything less mistakes will.

Reception third . If your equation has fractional coefficients, get rid of the fractions! Multiply the equation by the common denominator as described in the lesson "How to solve equations? Identity transformations". When working with fractions, errors, for some reason, climb ...

By the way, I promised an evil example with a bunch of minuses to simplify. You are welcome! There he is.

In order not to get confused in the minuses, we multiply the equation by -1. We get:

That's all! Deciding is fun!

So let's recap the topic.

Practical Tips:

1. Before solving, we bring the quadratic equation to the standard form, build it right.

2. If there is a negative coefficient in front of the x in the square, we eliminate it by multiplying the entire equation by -1.

3. If the coefficients are fractional, we eliminate the fractions by multiplying the entire equation by the corresponding factor.

4. If x squared is clean, the coefficient at it equal to one, the solution can be easily verified by Vieta's theorem. Do it!

Now you can decide.)

Solve Equations:

8x 2 - 6x + 1 = 0

x 2 + 3x + 8 = 0

x 2 - 4x + 4 = 0

(x+1) 2 + x + 1 = (x+1)(x+2)

Answers (in disarray):

x 1 = 0
x 2 = 5

x 1.2 =2

x 1 = 2
x 2 \u003d -0.5

x - any number

x 1 = -3
x 2 = 3

no solutions

x 1 = 0.25
x 2 \u003d 0.5

Does everything fit? Fine! Quadratic equations are not yours headache. The first three turned out, but the rest did not? Then the problem is not in quadratic equations. The problem is in identical transformations of equations. Take a look at the link, it's helpful.

Doesn't quite work? Or does it not work at all? Then Section 555 will help you. There, all these examples are sorted by bones. Showing main errors in the solution. Of course, it also talks about the use identical transformations in solving various equations. Helps a lot!

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

”, that is, equations of the first degree. In this lesson, we will explore what is a quadratic equation and how to solve it.

What is a quadratic equation

Important!

The degree of an equation is determined by the highest degree to which the unknown stands.

If the maximum degree to which the unknown stands is “2”, then you have a quadratic equation.

Examples of quadratic equations

• 5x2 - 14x + 17 = 0
• −x 2 + x +

  1

  3

  = 0
• x2 + 0.25x = 0
• x 2 − 8 = 0

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

A x 2 + b x + c = 0

"a", "b" and "c" - given numbers.

• "a" - the first or senior coefficient;
• "b" - 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 \u003d 0".

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

5x2 - 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

x2 + 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 general view"ax 2 + bx + c = 0". That is, only "0" should remain on the right side;
• use the formula for roots:

Let's use an example to figure out how to apply the 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 need only apply 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 \u003d" the root expression is often replaced
"b 2 − 4ac" to the letter "D" and called 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

In this form, it is rather difficult to determine the coefficients "a", "b", and "c". Let's first bring the equation to the general form "ax 2 + bx + c \u003d 0".

X 2 + 9 + x = 7x
x 2 + 9 + x − 7x = 0
x2 + 9 - 6x = 0
x 2 − 6x + 9 = 0

Now you can use the formula for the roots.

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 appears in the formula under the root.