Quadratic equations are used in solving many problems. A significant part of the problems that are easily solved with the help of equations of the first degree can also be solved purely arithmetically, although sometimes in a much more difficult, lengthy and often artificial way. Problems that lead to quadratic equations, as a rule, do not lend themselves to arithmetic solution at all. Numerous and most varied questions of physics, mechanics, hydromechanics, aerodynamics and many other applied sciences lead to such problems.

The main stages of compiling quadratic equations according to the conditions of the problem are the same as in solving problems leading to equations of the first degree. Let's give examples.

Task. 1. Two typists retyped the manuscript in 6 hours. 40 min. At what time could each typist, working alone, retype the manuscript if the first one spent 3 hours more on this work than the second?

Decision. Let the second typist spend x hours reprinting the manuscript. This means that the first typist will spend hours on the same job.

We will find out what part of the whole work each typist performs in one hour and what part - both together.

The first typist completes a part in an hour

Second part.

Both typists perform a part.

Hence we have:

According to the meaning of the problem, a positive number

Multiply both sides of the equation by After simplification, we get a quadratic equation:

Since , the equation has two roots. By formula (B) we find:

But as it should be, that value is not valid for this task.

Answer. The first typist will spend hours on work, the second 12 hours.

Problem 2. The own speed of the aircraft km per hour. The plane flew a distance of 1 km twice: first downwind, then against the wind, and on the second flight it spent more hours. Calculate wind speed.

We will depict the course of the solution in the form of a diagram.

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.

SQUARE TRIPON III

§ 50 Quadratic equations

Equations of the form

ax 2 + bx+c = 0, (1)

where X- unknown value, a, b, c- given numbers ( a =/= 0) are called square.

Singling out the full square on the left side of the quadratic equation (see formula (1) § 49), we obtain:

Obviously, equation (2) is equivalent to equation (1) (see § 2). Equation (2) can only have real roots when or b 2 - 4ace > 0 (since 4 a 2 > 0).

In view of the special role played by the expression D = b 2 - 4ace when solving equation (1), this expression is given a special name - discriminant quadratic equation ax 2 + bx+c = 0 (or the discriminant of the square trinomial ax 2 + bx+c ). So, if the discriminant of a quadratic equation is negative, then the equation has no real roots.

If D = b 2 - 4ace > 0, then from (2) we get:

If the discriminant of a quadratic equation is non-negative, then this equation has real roots. They are written as a fraction, the numerator of which is the coefficient of the equation for X , taken with the opposite sign, plus or minus the square root of the discriminant, and in the denominator - twice the coefficient at X 2 .

If the discriminant of a quadratic equation is positive, then the equation has two different real roots:

If the discriminant of a quadratic equation is zero, then the equation has one real root:

X = - b / 2 a

(In this case, the equation is sometimes said to have two equal roots: x 1 = x 2 = - b / 2 a )

Examples.

1) For Equation 2 X 2 - X - 3 = 0 discriminant D = (- 1) 2 - 4 2 (- 3) = 25 > 0. The equation has two different roots:

2) For Equation 3 X 2 - 6X + 3 = 0 D = (- 6) 2 - 4 3 3 = 0. This equation has one real root

3) For Equation 5 X 2 + 4X + 7 = 0 D = 4 2 - 4 5 7 = - 124< 0. Это уравнение не имеет действительных корней.

4) Find out for what values a quadratic equation X 2 + Oh + 1 = 0:

a) has one root

b) has two different roots;

c) has no roots at all,

The discriminant of this quadratic equation is

D= a 2 - 4.

If a | a | = 2, then D = 0; in this case, the equation has one root.

If a | a | > 2, then D > 0; in this case, the equation has two different roots.

Finally, if | a | < 2, то данное уравнение не имеет корней.

Exercises

Solve Equations (No. 364-369):

364. 6X 2 - X - 1 = 0. 367. - X 2 + 8X - 16 = 0.

365. 3X 2 - 5X + 1 = 0. 368. 2X 2 - 12X + 12 == 0.

366. X 2 - X + 1 = 0. 369. 2X - X 2 - 6 = 0.

370. Can the number 15 be represented as the sum of two numbers so that their product is equal to 70?

371. At what values a the equation

X 2 - 2Oh + a (1 + a ) = 0

a) has two different roots;

b) has only one root;

c) has no roots?

372. At what values a the equation

(1 - a ) X 2 - 4Oh + 4 (1 - a ) = 0

a) has no roots;

b) has no more than one root;

c) has at least one root?

373. At what value a the equation X 2 + Oh + 1 = 0 has a unique root? What is it equal to?

374. What are the limits of the number a , if it is known that the equations

X 2 + x + a = 0 and X 2 + x - a = 0

375. What can you say about the size a if the equations

4a (X 2 + X ) = a - 2.5 and X (X - 1) = 1,25 - a

have the same number of roots?

376. The train was delayed at the station for t min. To make up for lost time, the driver increased his speed by a km/h and on the next stage in b km eliminated the delay. How fast was the train before the delay at the station?

377. Two cranes, working together, unloaded the barge for t h. In what time can each crane separately unload the barge, if one of them spends on it for a h less than the other?

378. One of the factories fulfills some order 4 days faster than the other. How long can each plant complete an order, working separately, if it is known that when working together in 24 days they completed an order 5 times larger?

Solve equations (No. 379, 380).

(Pay attention to the fact that in these equations the unknown is contained in the denominators of fractions. The resulting roots will need to be checked!)

381*. At what values a equations

X 2 + Oh + 1 = 0 and X 2 + X + a = 0

have at least one common root?

Farafonova Natalia Igorevna

Subject: Incomplete quadratic equations.

Lesson Objectives:- Introduce the concept of an incomplete quadratic equation;

Learn how to solve incomplete quadratic equations.

Lesson objectives:- Be able to determine the form of a quadratic equation;

Solve incomplete quadratic equations.

Webbook: Algebra: Proc. for 8 cells. general education institutions / Sh. A. Alimov, Yu. M. Kolyagin, Yu. V. Sidorov and others - M .: Education, 2010.

During the classes.

1. Remind students that before solving any quadratic equation, it is necessary to bring it to a standard form. Remember the definition full quadratic equation:ax2+bx +c = 0,a ≠ 0.

In these quadratic equations, name the coefficients a, b, c:

a) 2x 2 - x + 3 = 0; b) x 2 + 4x - 1 = 0; c) x 2 - 4 \u003d 0; d) 5x 2 + 3x = 0.

2. Give a definition of an incomplete quadratic equation:

The quadratic equation ax 2 + bx + c = 0 is called incomplete, if at least one of the coefficients, b or c, is equal to 0. Pay attention that the coefficient a ≠ 0. From the equations presented above, choose incomplete quadratic equations.

3. It is more convenient to present the types of incomplete quadratic equations with examples of solutions in the form of a table:

1. Without solving, determine the number of roots for each incomplete quadratic equation:

a) 2x 2 - 3 = 0; b) 3x 2 + 4 = 0; c) 5x 2 - x \u003d 0; d) 0.6x2 = 0; e) -8x 2 - 4 = 0.

1. Solve incomplete quadratic equations (solution of equations, with a check at the blackboard, 2 options):

c) 2x 2 + 15 = 0

d) 3x 2 + 2x = 0

e) 2x 2 - 16 = 0

f) 5(x 2 + 2) = 2(x 2 + 5)

g) (x + 1) 2 - 4 = 0

c) 2x 2 + 7 = 0

d) x 2 + 9x = 0

e) 81x 2 - 64 = 0

f) 2(x 2 + 4) = 4(x 2 + 2)

g) (x - 2) 2 - 8 = 0.

6. Independent work on options:

1 option

a) 3x 2 - 12 = 0

b) 2x 2 + 6x = 0

e) 7x 2 - 14 = 0

Option 2

b) 6x 2 + 24 = 0

c) 9y 2 - 4 = 0

d) -y 2 + 5 = 0

e) 1 - 4y 2 = 0

f) 8y 2 + y = 0

3 option

a) 6y - y 2 = 0

b) 0.1y 2 - 0.5y = 0

c) (x + 1) (x -2) = 0

d) x(x + 0.5) = 0

e) x 2 - 2x = 0

f) x 2 - 16 = 0

4 option

a) 9x 2 - 1 = 0

b) 3x - 2x 2 = 0

d) x 2 + 2x - 3 = 2x + 6

e) 3x 2 + 7 = 12x + 7

5 option

a) 2x 2 - 18 = 0

b) 3x 2 - 12x = 0

d) x 2 + 16 = 0

e) 6x 2 - 18 = 0

f) x 2 - 5x = 0

6 option

b) 4x 2 + 36 = 0

c) 25y 2 - 1 = 0

d) -y 2 + 2 = 0

e) 9 - 16y 2 = 0

f) 7y 2 + y = 0

7 option

a) 4y - y 2 = 0

b) 0.2y 2 - y = 0

c) (x + 2)(x - 1) = 0

d) (x - 0.3)x = 0

e) x 2 + 4x = 0

f) x 2 - 36 = 0

8 option

a) 16x 2 - 1 = 0

b) 4x - 5x 2 = 0

d) x 2 - 3x - 5 = 11 - 3x

e) 5x 2 - 6 = 15x - 6

Answers for independent work:

Option 1: a) 2, b) 0; -3; c) 0; d) there are no roots; e);

Option 2 a) 0; b) roots; in); G); e); f)0;-;

3 option a) 0; 6; b) 0;5; c) -1;2; d) 0; -0.5; e) 0;2; f)4

4 option a); b) 0; 1.5; c) 0;3; d) 3; e)0;4 e)5

5 option a)3; b) 0;4; c) 0; d) there are no roots; e) f) 0; 5

6 option a) 0; b) there are no roots; c) d) e) f) 0;-

7 option a) 0; 4; b) 0;5; c) -2;1; d) 0; 0.03; e) 0;-4; f)6

8 option a) b) 0; c) 0;7; d) 4; e) 0;3; e)

Lesson summary: The concept of "incomplete quadratic equation" is formulated; ways of solving different types of incomplete quadratic equations are shown. In the course of performing various tasks, the skills of solving incomplete quadratic equations were worked out.

7. Homework: №№ 421(2), 422(2), 423(2,4), 425.

Additional task:

For what values ​​of a is the equation an incomplete quadratic equation? Solve the equation for the obtained values ​​of a:

a) x 2 + 3ax + a - 1 = 0

b) (a - 2)x 2 + ax \u003d 4 - a 2 \u003d 0