In this article, we will consider the solution of incomplete quadratic equations.
But first, let's repeat what equations are called quadratic. An equation of the form ax 2 + bx + c \u003d 0, where x is a variable, and the coefficients a, b and c are some numbers, and a ≠ 0, is called square. As we can see, the coefficient at x 2 is not equal to zero, and therefore the coefficients at x or the free term can be equal to zero, in this case we get an incomplete quadratic equation.
There are three kinds of incomplete quadratic equations:
1) If b \u003d 0, c ≠ 0, then ax 2 + c \u003d 0;
2) If b ≠ 0, c \u003d 0, then ax 2 + bx \u003d 0;
3) If b \u003d 0, c \u003d 0, then ax 2 \u003d 0.
- Let's see how they solve equations of the form ax 2 + c = 0.
To solve the equation, we transfer the free term from to the right side of the equation, we get
ax 2 = ‒s. Since a ≠ 0, then we divide both parts of the equation by a, then x 2 \u003d -c / a.
If ‒с/а > 0, then the equation has two roots
x = ±√(–c/a) .
If ‒c/a< 0, то это уравнение решений не имеет. Более наглядно решение данных уравнений представлено на схеме.
Let's try to understand with examples how to solve such equations.
Example 1. Solve the equation 2x 2 - 32 = 0.
Answer: x 1 \u003d - 4, x 2 \u003d 4.
Example 2. Solve the equation 2x 2 + 8 = 0.
Answer: The equation has no solutions.
- Let's see how they solve equations of the form ax 2 + bx = 0.
To solve the equation ax 2 + bx \u003d 0, we decompose it into factors, that is, we take x out of brackets, we get x (ax + b) \u003d 0. The product is zero if at least one of the factors is zero. Then either х = 0 or ах + b = 0. Solving the equation ах + b = 0, we obtain ах = – b, whence х = – b/a. An equation of the form ax 2 + bx \u003d 0 always has two roots x 1 \u003d 0 and x 2 \u003d - b / a. See how the solution of equations of this type looks on the diagram. 
Let's consolidate our knowledge on a specific example.
Example 3. Solve the equation 3x 2 - 12x = 0.
x(3x - 12) = 0
x \u003d 0 or 3x - 12 \u003d 0
Answer: x 1 = 0, x 2 = 4.
- Equations of the third type ax 2 = 0 solved very simply.
If ax 2 \u003d 0, then x 2 \u003d 0. The equation has two equal root x 1 = 0, x 2 = 0.
For clarity, consider the diagram. 
When solving Example 4, we will make sure that equations of this type are solved very simply.
Example 4 Solve the equation 7x 2 = 0.
Answer: x 1, 2 = 0.
It is not always immediately clear what kind of incomplete quadratic equation we have to solve. Consider the following example.
Example 5 solve the equation
Multiply both sides of the equation by common denominator, that is, 30
Let's cut
5 (5x 2 + 9) - 6 (4x 2 - 9) \u003d 90.
Let's open the brackets
25x2 + 45 - 24x2 + 54 = 90.
Here are similar
Let's move 99 from the left side of the equation to the right, changing the sign to the opposite
Answer: no roots.
We have analyzed how incomplete quadratic equations are solved. I hope now you will not have difficulties with such tasks. Be careful when determining the type of an incomplete quadratic equation, then you will succeed.
If you have any questions on this topic, sign up for my lessons, we will solve the problems together.
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Quadratic equations. General information.
IN quadratic equation there must be an x in the square (that's why it is called
"square"). 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.
Algebraic equation general view.
Where x is a free variable, a, b, c are coefficients, and a≠0 .
For example: 
Expression 
called square trinomial.
The elements of a quadratic equation have their own names:
called the first or senior coefficient,
is called the second or coefficient at ,
is called a free member.
Complete quadratic equation.
These quadratic equations have the full set of terms on the left. x squared
coefficient A, x to the first power with coefficient b And free memberWith. IN all coefficients
must be different from zero.
Incomplete is a quadratic equation in which at least one of the coefficients, except for
senior (either the second coefficient or the free term) is equal to zero.
Let's pretend that b\u003d 0, - x will disappear in the first degree. It turns out, for example:
2x 2 -6x=0,
And so on. And if both coefficients b And c are equal to zero, then it is even simpler, For example:
2x 2 \u003d 0,
Note that x squared is present in all equations.
Why A can't be zero? Then the x squared disappears and the equation becomes linear .
And it's done differently...
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
And so on. 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. According to formulas and clear simple rules. The first step is to bring the given equation 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 be confused?), But with the substitution of 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 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.
All. 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 by simply transferring the number to the right, followed by extracting the root.
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 to use.) I remind you of the most general formula for solving 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. From negative number the square root is not taken. 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 complex 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! There will be fewer mistakes.
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. Please! Here 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.
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 pure, the coefficient for it is equal to one, the solution can be easily checked 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? Great! 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.
Quadratic equation or an equation of the second degree with one unknown is an equation that, after transformations, can be reduced to the following form:
ax 2 + bx + c = 0 - quadratic equation
Where x is the unknown, and a, b And c- coefficients of the equation. In quadratic equations a is called the first coefficient ( a ≠ 0), b is called the second coefficient, and c is called a known or free member.
The equation:
ax 2 + bx + c = 0
called complete quadratic equation. If one of the coefficients b or c is zero, or both of these coefficients are equal to zero, then the equation is presented as an incomplete quadratic equation.
Reduced quadratic equation
The complete quadratic equation can be reduced to a more convenient form by dividing all its terms by a, that is, for the first coefficient:
The equation x 2 + px + q= 0 is called a reduced quadratic equation. Therefore, any quadratic equation in which the first coefficient is equal to 1 can be called reduced.
For example, the equation:
x 2 + 10x - 5 = 0
is reduced, and the equation:
3x 2 + 9x - 12 = 0
can be replaced by the above equation by dividing all its terms by -3:
x 2 - 3x + 4 = 0
Solving quadratic equations
To solve a quadratic equation, you need to bring it to one of the following forms:
ax 2 + bx + c = 0
ax 2 + 2kx + c = 0
x 2 + px + q = 0
Each type of equation has its own formula for finding the roots:
Pay attention to the equation:
ax 2 + 2kx + c = 0
this is the converted equation ax 2 + bx + c= 0, in which the coefficient b- even, which allows it to be replaced by type 2 k. Therefore, the formula for finding the roots for this equation can be simplified by substituting 2 k instead of b:
Example 1 Solve the equation:
3x 2 + 7x + 2 = 0
Since in the equation the second coefficient is not an even number, and the first coefficient is not equal to one, we will look for the roots using the very first formula, called general formula finding the roots of a quadratic equation. At first
a = 3, b = 7, c = 2
Now, to find the roots of the equation, we simply substitute the values of the coefficients into the formula:
| x 1 = | -2 | = - | 1 | , x 2 = | -12 | = -2 |
| 6 | 3 | 6 |
| Answer: - | 1 | , -2. |
| 3 |
Example 2:
x 2 - 4x - 60 = 0
Let's determine what the coefficients are equal to:
a = 1, b = -4, c = -60
Since the second coefficient in the equation is an even number, we will use the formula for quadratic equations with an even second coefficient:
x 1 = 2 + 8 = 10, x 2 = 2 - 8 = -6
Answer: 10, -6.
Example 3
y 2 + 11y = y - 25
Let's bring the equation to general view:
y 2 + 11y = y - 25
y 2 + 11y - y + 25 = 0
y 2 + 10y + 25 = 0
Let's determine what the coefficients are equal to:
a = 1, p = 10, q = 25
Since the first coefficient is equal to 1, we will look for the roots using the formula for the above equations with an even second coefficient:
Answer: -5.
Example 4
x 2 - 7x + 6 = 0
Let's determine what the coefficients are equal to:
a = 1, p = -7, q = 6
Since the first coefficient is equal to 1, we will look for the roots using the formula for the given equations with an odd second coefficient:
x 1 = (7 + 5) : 2 = 6, x 2 = (7 - 5) : 2 = 1
Solving equations using the "transfer" method
Consider the quadratic equation
ax 2 + bx + c \u003d 0, where a? 0.
Multiplying both its parts by a, we obtain the equation
a 2 x 2 + abx + ac = 0.
Let ax = y, whence x = y/a; then we come to the equation
y 2 + by + ac = 0,
equivalent to this one. We find its roots at 1 and at 2 using the Vieta theorem.
Finally we get x 1 = y 1 /a and x 1 = y 2 /a. With this method, the coefficient a is multiplied by the free term, as if “transferred” to it, therefore it is called the “transfer” method. This method is used when it is easy to find the roots of an equation using Vieta's theorem and, most importantly, when the discriminant is an exact square.
* Example.
We solve the equation 2x 2 - 11x + 15 = 0.
Solution. Let's "transfer" the coefficient 2 to the free term, as a result we get the equation
y 2 - 11y + 30 = 0.
According to Vieta's theorem
y 1 = 5 x 1 = 5/2 x 1 = 2.5
y 2 = 6 x 2 = 6/2 x 2 = 3.
Answer: 2.5; 3.
Properties of the coefficients of a quadratic equation
A. Let a quadratic equation ax 2 + bx + c = 0 be given, where a? 0.
1) If, a + b + c \u003d 0 (i.e., the sum of the coefficients is zero), then x 1 \u003d 1,
Proof. Divide both sides of the equation by a? 0, we get the reduced quadratic equation
x 2 + b/a * x + c/a = 0.
According to Vieta's theorem
x 1 + x 2 \u003d - b / a,
x 1 x 2 = 1*c/a.
By condition a - b + c = 0, whence b = a + c. Thus,
x 1 + x 2 \u003d - a + b / a \u003d -1 - c / a,
x 1 x 2 \u003d - 1 * (- c / a),
those. x 1 \u003d -1 and x 2 \u003d c / a, which m was required to prove.
- * Examples.
- 1) Let's solve the equation 345x 2 - 137x - 208 = 0.
Solution. Since a + b + c \u003d 0 (345 - 137 - 208 \u003d 0), then
x 1 = 1, x 2 = c / a = -208/345.
Answer: 1; -208/345.
2) Solve the equation 132x 2 - 247x + 115 = 0.
Solution. Since a + b + c \u003d 0 (132 - 247 + 115 \u003d 0), then
x 1 \u003d 1, x 2 \u003d c / a \u003d 115/132.
Answer: 1; 115/132.
B. If the second coefficient b = 2k is an even number, then the root formula
* Example.
Let's solve the equation 3x2 - 14x + 16 = 0.
Solution. We have: a = 3, b = - 14, c = 16, k = - 7;