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 much.

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.

It is known that it is a particular version of the equality ax 2 + in + c \u003d o, where a, b and c are real coefficients for unknown x, and where a ≠ o, and b and c will be zeros - simultaneously or separately. For example, c = o, v ≠ o or vice versa. We almost remembered the definition of a quadratic equation.

A trinomial of the second degree is equal to zero. Its first coefficient a ≠ o, b and c can take on any values. The value of the variable x will then be when, when substituting, it will turn it into the correct numerical equality. Let us dwell on real roots, although the solutions of the equation can also be complete. It is customary to call an equation in which none of the coefficients is equal to o, a ≠ o, b ≠ o, c ≠ o.
Let's solve an example. 2x2 -9x-5 = oh, we find
D \u003d 81 + 40 \u003d 121,
D is positive, so there are roots, x 1 = (9+√121): 4 = 5, and the second x 2 = (9-√121): 4 = -o.5. Checking will help make sure they are correct.

Here is a step by step solution to the quadratic equation

Through the discriminant, you can solve any equation, on the left side of which there is a known square trinomial with a ≠ o. In our example. 2x 2 -9x-5 \u003d 0 (ax 2 + in + c \u003d o)

Consider what are incomplete equations of the second degree

1. ax 2 + in = o. The free term, the coefficient c at x 0, is zero here, in ≠ o.
   How to solve an incomplete quadratic equation of this kind? Let's take x out of brackets. Remember when the product of two factors is zero.
   x(ax+b) = o, this can be when x = o or when ax+b = o.
   Solving the 2nd, we have x = -v/a.
   As a result, we have roots x 1 \u003d 0, according to calculations x 2 \u003d -b / a.
2. Now the coefficient of x is o, but c is not equal to (≠) o.
   x 2 + c \u003d o. We transfer c to the right side of the equality, we get x 2 \u003d -c. This equation only has real roots when -c is a positive number (c ‹ o),
   x 1 is then equal to √(-c), respectively, x 2 is -√(-c). Otherwise, the equation has no roots at all.
3. The last option: b \u003d c \u003d o, that is, ax 2 \u003d o. Naturally, such a simple equation has one root, x = o.

Special cases

We have considered how to solve an incomplete quadratic equation, and now we will take any kind.

• In the full quadratic equation, the second coefficient of x is an even number.
  Let k = o,5b. We have formulas for calculating the discriminant and roots.
  D / 4 \u003d k 2 - ac, the roots are calculated as follows x 1,2 \u003d (-k ± √ (D / 4)) / a for D › o.
  x = -k/a at D = o.
  There are no roots for D ‹ o.
• There are reduced quadratic equations, when the coefficient of x squared is 1, they are usually written x 2 + px + q \u003d o. All of the above formulas apply to them, but the calculations are somewhat simpler.
  Example, x 2 -4x-9 \u003d 0. We calculate D: 2 2 +9, D \u003d 13.
  x 1 = 2+√13, x 2 = 2-√13.
• In addition, it is easy to apply to the given ones. It says that the sum of the roots of the equation is equal to -p, the second coefficient with a minus (meaning the opposite sign), and the product of these same roots will be equal to q, the free term. Check out how easy it would be to verbally determine the roots of this equation. For non-reduced (for all coefficients that are not equal to zero), this theorem is applicable as follows: the sum x 1 + x 2 is equal to -v / a, the product x 1 x 2 is equal to c / a.

The sum of the free term c and the first coefficient a is equal to the coefficient b. In this situation, the equation has at least one root (it is easy to prove), the first one is necessarily equal to -1, and the second - c / a, if it exists. How to solve an incomplete quadratic equation, you can check it yourself. As easy as pie. Coefficients can be in some ratios among themselves

• x 2 + x \u003d o, 7x 2 -7 \u003d o.
• The sum of all coefficients is o.
  The roots of such an equation are 1 and c / a. Example, 2x 2 -15x + 13 = o.
  x 1 \u003d 1, x 2 \u003d 13/2.

There are a number of other ways to solve different equations of the second degree. Here, for example, is a method for extracting a full square from a given polynomial. There are several graphic ways. When you often deal with such examples, you will learn to “click” them like seeds, because all methods come to mind automatically.

With this math program you can solve quadratic equation.

The program not only gives the answer to the problem, but also displays the solution process in two ways:
- using the discriminant
- using the Vieta theorem (if possible).

Moreover, the answer is displayed exact, not approximate.
For example, for the equation \(81x^2-16x-1=0\), the answer is displayed in this form:

$$ x_1 = \frac(8+\sqrt(145))(81), \quad x_2 = \frac(8-\sqrt(145))(81) $$ instead of this: \(x_1 = 0.247; \quad x_2 = -0.05 \)

This program can be useful for high school students in preparation for tests and exams, when testing knowledge before the Unified State Exam, for parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with a detailed solution.

In this way, you can conduct your own training and/or the training of your younger brothers or sisters, while the level of education in the field of tasks to be solved is increased.

If you are not familiar with the rules for entering a square polynomial, we recommend that you familiarize yourself with them.

Rules for entering a square polynomial

Any Latin letter can act as a variable.
For example: \(x, y, z, a, b, c, o, p, q \) etc.

Numbers can be entered as integers or fractions.
Moreover, fractional numbers can be entered not only in the form of a decimal, but also in the form of an ordinary fraction.

Rules for entering decimal fractions.
In decimal fractions, the fractional part from the integer can be separated by either a dot or a comma.
For example, you can enter decimals like this: 2.5x - 3.5x^2

Rules for entering ordinary fractions.
Only a whole number can act as the numerator, denominator and integer part of a fraction.

The denominator cannot be negative.

When entering a numerical fraction, the numerator is separated from the denominator by a division sign: /
The integer part is separated from the fraction by an ampersand: &
Input: 3&1/3 - 5&6/5z +1/7z^2
Result: \(3\frac(1)(3) - 5\frac(6)(5) z + \frac(1)(7)z^2 \)

When entering an expression you can use brackets. In this case, when solving a quadratic equation, the introduced expression is first simplified.
For example: 1/2(y-1)(y+1)-(5y-10&1/2)

=0

Decide

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A bit of theory.

Quadratic equation and its roots. Incomplete quadratic equations

Each of the equations
\(-x^2+6x+1,4=0, \quad 8x^2-7x=0, \quad x^2-\frac(4)(9)=0 \)
has the form
\(ax^2+bx+c=0, \)
where x is a variable, a, b and c are numbers.
In the first equation a = -1, b = 6 and c = 1.4, in the second a = 8, b = -7 and c = 0, in the third a = 1, b = 0 and c = 4/9. Such equations are called quadratic equations.

Definition.
quadratic equation an equation of the form ax 2 +bx+c=0 is called, where x is a variable, a, b and c are some numbers, and \(a \neq 0 \).

The numbers a, b and c are the coefficients of the quadratic equation. The number a is called the first coefficient, the number b is the second coefficient and the number c is the intercept.

In each of the equations of the form ax 2 +bx+c=0, where \(a \neq 0 \), the largest power of the variable x is a square. Hence the name: quadratic equation.

Note that a quadratic equation is also called an equation of the second degree, since its left side is a polynomial of the second degree.

A quadratic equation in which the coefficient at x 2 is 1 is called reduced quadratic equation. For example, the given quadratic equations are the equations
\(x^2-11x+30=0, \quad x^2-6x=0, \quad x^2-8=0 \)

If in the quadratic equation ax 2 +bx+c=0 at least one of the coefficients b or c is equal to zero, then such an equation is called incomplete quadratic equation. So, the equations -2x 2 +7=0, 3x 2 -10x=0, -4x 2 =0 are incomplete quadratic equations. In the first of them b=0, in the second c=0, in the third b=0 and c=0.

Incomplete quadratic equations are of three types:
1) ax 2 +c=0, where \(c \neq 0 \);
2) ax 2 +bx=0, where \(b \neq 0 \);
3) ax2=0.

Consider the solution of equations of each of these types.

To solve an incomplete quadratic equation of the form ax 2 +c=0 for \(c \neq 0 \), its free term is transferred to the right side and both parts of the equation are divided by a:
\(x^2 = -\frac(c)(a) \Rightarrow x_(1,2) = \pm \sqrt( -\frac(c)(a)) \)

Since \(c \neq 0 \), then \(-\frac(c)(a) \neq 0 \)

If \(-\frac(c)(a)>0 \), then the equation has two roots.

If \(-\frac(c)(a) To solve an incomplete quadratic equation of the form ax 2 +bx=0 for \(b \neq 0 \) factorize its left side and obtain the equation
\(x(ax+b)=0 \Rightarrow \left\( \begin(array)(l) x=0 \\ ax+b=0 \end(array) \right. \Rightarrow \left\( \begin (array)(l) x=0 \\ x=-\frac(b)(a) \end(array) \right. \)

Hence, an incomplete quadratic equation of the form ax 2 +bx=0 for \(b \neq 0 \) always has two roots.

An incomplete quadratic equation of the form ax 2 \u003d 0 is equivalent to the equation x 2 \u003d 0 and therefore has a single root 0.

The formula for the roots of a quadratic equation

Let us now consider how quadratic equations are solved in which both coefficients of the unknowns and the free term are nonzero.

We solve the quadratic equation in general form and as a result we obtain the formula of the roots. Then this formula can be applied to solve any quadratic equation.

Solve the quadratic equation ax 2 +bx+c=0

Dividing both its parts by a, we obtain the equivalent reduced quadratic equation
\(x^2+\frac(b)(a)x +\frac(c)(a)=0 \)

We transform this equation by highlighting the square of the binomial:
\(x^2+2x \cdot \frac(b)(2a)+\left(\frac(b)(2a)\right)^2- \left(\frac(b)(2a)\right)^ 2 + \frac(c)(a) = 0 \Rightarrow \)

\(x^2+2x \cdot \frac(b)(2a)+\left(\frac(b)(2a)\right)^2 = \left(\frac(b)(2a)\right)^ 2 - \frac(c)(a) \Rightarrow \) \(\left(x+\frac(b)(2a)\right)^2 = \frac(b^2)(4a^2) - \frac( c)(a) \Rightarrow \left(x+\frac(b)(2a)\right)^2 = \frac(b^2-4ac)(4a^2) \Rightarrow \) \(x+\frac(b )(2a) = \pm \sqrt( \frac(b^2-4ac)(4a^2) ) \Rightarrow x = -\frac(b)(2a) + \frac( \pm \sqrt(b^2 -4ac) )(2a) \Rightarrow \) \(x = \frac( -b \pm \sqrt(b^2-4ac) )(2a) \)

The root expression is called discriminant of a quadratic equation ax 2 +bx+c=0 (“discriminant” in Latin - distinguisher). It is denoted by the letter D, i.e.
\(D = b^2-4ac\)

Now, using the notation of the discriminant, we rewrite the formula for the roots of the quadratic equation:
\(x_(1,2) = \frac( -b \pm \sqrt(D) )(2a) \), where \(D= b^2-4ac \)

It's obvious that:
1) If D>0, then the quadratic equation has two roots.
2) If D=0, then the quadratic equation has one root \(x=-\frac(b)(2a)\).
3) If D Thus, depending on the value of the discriminant, the quadratic equation can have two roots (for D > 0), one root (for D = 0) or no roots (for D When solving a quadratic equation using this formula, it is advisable to do the following way:
1) calculate the discriminant and compare it with zero;
2) if the discriminant is positive or equal to zero, then use the root formula, if the discriminant is negative, then write down that there are no roots.

Vieta's theorem

The given quadratic equation ax 2 -7x+10=0 has roots 2 and 5. The sum of the roots is 7, and the product is 10. We see that the sum of the roots is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term. Any reduced quadratic equation that has roots has this property.

The sum of the roots of the given quadratic equation is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term.

Those. Vieta's theorem states that the roots x 1 and x 2 of the reduced quadratic equation x 2 +px+q=0 have the property:
\(\left\( \begin(array)(l) x_1+x_2=-p \\ x_1 \cdot x_2=q \end(array) \right. \)

The equation takes the form:

Let's solve it in general form:

Comment: the equation will have roots only if , otherwiseturns out to be a square

equals a negative number, which is impossible.

Answer:

Example:

Answer:

The last transition was made because the irrationality in the denominator is left extremely rarely.

2. Free member is zero(c=0).

The equation takes the form:

Let's solve it in general form:

For solutions given quadratic equations, i.e. if the coefficient

a= 1:

x2+bx+c=0,

then x 1 x 2 =c

x 1 +x 2 =−b

For a complete quadratic equation in which a≠1:

x2+bx+c=0,

divide the whole equation by a:

→ →

where x 1 and x 2 - roots of the equation.

Reception third. If your equation has fractional coefficients, get rid offractions! Multiply

the equation to a common denominator.

Conclusion. 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 multiplication

of the whole equation by -1.

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

factor.

4. If x squared is pure, the coefficient with it is equal to one, the solution can be easily check by

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