Farafonova Natalia Igorevna
Topic: 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 complete 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:
- 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.
- 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
Tasks for a quadratic equation are studied both in the 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
The root of this value is 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 
Using well-known formulas, we 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 - the larger 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 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.
First level
Quadratic equations. Comprehensive Guide (2019)
In the term "quadratic equation" the key word is "quadratic". This means that the equation must necessarily contain a variable (the same X) in the square, and at the same time there should not be Xs in the third (or greater) degree.
The solution of many equations is reduced to the solution of quadratic equations.
Let's learn to determine that we have a quadratic equation, and not some other.
Example 1
Get rid of the denominator and multiply each term of the equation by
Let's move everything to the left side and arrange the terms in descending order of powers of x
Now we can say with confidence that this equation is quadratic!
Example 2
Multiply the left and right sides by:
This equation, although it was originally in it, is not a square!
Example 3
Let's multiply everything by:
Scary? The fourth and second degrees ... However, if we make a replacement, we will see that we have a simple quadratic equation:
Example 4
It seems to be, but let's take a closer look. Let's move everything to the left side:
You see, it has shrunk - and now it's a simple linear equation!
Now try to determine for yourself which of the following equations are quadratic and which are not:
Examples:
Answers:
- square;
- square;
- not square;
- not square;
- not square;
- square;
- not square;
- square.
Mathematicians conditionally divide all quadratic equations into the following types:
- Complete quadratic equations- equations in which the coefficients and, as well as the free term c, are not equal to zero (as in the example). In addition, among the complete quadratic equations, there are given are equations in which the coefficient (the equation from example one is not only complete, but also reduced!)
- Incomplete quadratic equations- equations in which the coefficient and or free term c are equal to zero:
They are incomplete because some element is missing from them. But the equation must always contain x squared !!! Otherwise, it will no longer be a quadratic, but some other equation.
Why did they come up with such a division? It would seem that there is an X squared, and okay. Such a division is due to the methods of solution. Let's consider each of them in more detail.
Solving incomplete quadratic equations
First, let's focus on solving incomplete quadratic equations - they are much simpler!
Incomplete quadratic equations are of types:
- , in this equation the coefficient is equal.
- , in this equation the free term is equal to.
- , in this equation the coefficient and the free term are equal.
1. i. Since we know how to take the square root, let's express from this equation
The expression can be either negative or positive. A squared number cannot be negative, because when multiplying two negative or two positive numbers, the result will always be a positive number, so: if, then the equation has no solutions.
And if, then we get two roots. These formulas do not need to be memorized. The main thing is that you should always know and remember that it cannot be less.
Let's try to solve some examples.
Example 5:
Solve the Equation
Now it remains to extract the root from the left and right parts. After all, do you remember how to extract the roots?
Answer:
Never forget about roots with a negative sign!!!
Example 6:
Solve the Equation
Answer:
Example 7:
Solve the Equation
Ouch! The square of a number cannot be negative, which means that the equation
no roots!
For such equations in which there are no roots, mathematicians came up with a special icon - (empty set). And the answer can be written like this:
Answer:
Thus, this quadratic equation has two roots. There are no restrictions here, since we did not extract the root.
Example 8:
Solve the Equation
Let's take the common factor out of brackets:
In this way,
This equation has two roots.
Answer:
The simplest type of incomplete quadratic equations (although they are all simple, right?). Obviously, this equation always has only one root:
Here we will do without examples.
Solving complete quadratic equations
We remind you that the complete quadratic equation is an equation of the form equation where
Solving full quadratic equations is a bit more complicated (just a little bit) than those given.
Remember, any quadratic equation can be solved using the discriminant! Even incomplete.
The rest of the methods will help you do it faster, but if you have problems with quadratic equations, first master the solution using the discriminant.
1. Solving quadratic equations using the discriminant.
Solving quadratic equations in this way is very simple, the main thing is to remember the sequence of actions and a couple of formulas.
If, then the equation has a root. Special attention should be paid to the step. The discriminant () tells us the number of roots of the equation.
- If, then the formula at the step will be reduced to. Thus, the equation will have only a root.
- If, then we will not be able to extract the root of the discriminant at the step. This indicates that the equation has no roots.
Let's go back to our equations and look at a few examples.
Example 9:
Solve the Equation
Step 1 skip.
Step 2
Finding the discriminant:
So the equation has two roots.
Step 3
Answer:
Example 10:
Solve the Equation
The equation is in standard form, so Step 1 skip.
Step 2
Finding the discriminant:
So the equation has one root.
Answer:
Example 11:
Solve the Equation
The equation is in standard form, so Step 1 skip.
Step 2
Finding the discriminant:
This means that we will not be able to extract the root from the discriminant. There are no roots of the equation.
Now we know how to write down such answers correctly.
Answer: no roots
2. Solution of quadratic equations using the Vieta theorem.
If you remember, then there is such a type of equations that are called reduced (when the coefficient a is equal to):
Such equations are very easy to solve using Vieta's theorem:
The sum of the roots given quadratic equation is equal, and the product of the roots is equal.
Example 12:
Solve the Equation
This equation is suitable for solution using Vieta's theorem, because .
The sum of the roots of the equation is, i.e. we get the first equation:
And the product is:
Let's create and solve the system:
- and. The sum is;
- and. The sum is;
- and. The amount is equal.
and are the solution of the system:
Answer: ; .
Example 13:
Solve the Equation
Answer:
Example 14:
Solve the Equation
The equation is reduced, which means:
Answer:
QUADRATIC EQUATIONS. AVERAGE LEVEL
What is a quadratic equation?
In other words, a quadratic equation is an equation of the form, where - unknown, - some numbers, moreover.
The number is called the highest or first coefficient quadratic equation, - second coefficient, a - free member.
Why? Because if, the equation will immediately become linear, because will disappear.
In this case, and can be equal to zero. In this stool equation is called incomplete. If all the terms are in place, that is, the equation is complete.
Solutions to various types of quadratic equations
Methods for solving incomplete quadratic equations:
To begin with, we will analyze the methods for solving incomplete quadratic equations - they are simpler.
The following types of equations can be distinguished:
I. , in this equation the coefficient and the free term are equal.
II. , in this equation the coefficient is equal.
III. , in this equation the free term is equal to.
Now consider the solution of each of these subtypes.
Obviously, this equation always has only one root:
A number squared cannot be negative, because when multiplying two negative or two positive numbers, the result will always be a positive number. That's why:
if, then the equation has no solutions;
if we have two roots
These formulas do not need to be memorized. The main thing to remember is that it cannot be less.
Examples:
Solutions:
Answer:
Never forget about roots with a negative sign!
The square of a number cannot be negative, which means that the equation
no roots.
To briefly write that the problem has no solutions, we use the empty set icon.
Answer:
So, this equation has two roots: and.
Answer:
Let's take the common factor out of brackets:
The product is equal to zero if at least one of the factors is equal to zero. This means that the equation has a solution when:
So, this quadratic equation has two roots: and.
Example:
Solve the equation.
Solution:
We factorize the left side of the equation and find the roots:
Answer:
Methods for solving complete quadratic equations:
1. Discriminant
Solving quadratic equations in this way is easy, the main thing is to remember the sequence of actions and a couple of formulas. Remember, any quadratic equation can be solved using the discriminant! Even incomplete.
Did you notice the root of the discriminant in the root formula? But the discriminant can be negative. What to do? We need to pay special attention to step 2. The discriminant tells us the number of roots of the equation.
- If, then the equation has a root:
- If, then the equation has the same root, but in fact, one root:
Such roots are called double roots.
- If, then the root of the discriminant is not extracted. This indicates that the equation has no roots.
Why are there different numbers of roots? Let us turn to the geometric meaning of the quadratic equation. The graph of the function is a parabola:
In a particular case, which is a quadratic equation, . And this means that the roots of the quadratic equation are the points of intersection with the x-axis (axis). The parabola may not cross the axis at all, or it may intersect it at one (when the top of the parabola lies on the axis) or two points.
In addition, the coefficient is responsible for the direction of the branches of the parabola. If, then the branches of the parabola are directed upwards, and if - then downwards.
Examples:
Solutions:
Answer:
Answer: .
Answer:
This means there are no solutions.
Answer: .
2. Vieta's theorem
Using the Vieta theorem is very easy: you just need to choose a pair of numbers whose product is equal to the free term of the equation, and the sum is equal to the second coefficient, taken with the opposite sign.
It is important to remember that Vieta's theorem can only be applied to given quadratic equations ().
Let's look at a few examples:
Example #1:
Solve the equation.
Solution:
This equation is suitable for solution using Vieta's theorem, because . Other coefficients: ; .
The sum of the roots of the equation is:
And the product is:
Let's select such pairs of numbers, the product of which is equal, and check if their sum is equal:
- and. The sum is;
- and. The sum is;
- and. The amount is equal.
and are the solution of the system:
Thus, and are the roots of our equation.
Answer: ; .
Example #2:
Solution:
We select such pairs of numbers that give in the product, and then check whether their sum is equal:
and: give in total.
and: give in total. To get it, you just need to change the signs of the alleged roots: and, after all, the product.
Answer:
Example #3:
Solution:
The free term of the equation is negative, and hence the product of the roots is a negative number. This is possible only if one of the roots is negative and the other is positive. So the sum of the roots is differences of their modules.
We select such pairs of numbers that give in the product, and the difference of which is equal to:
and: their difference is - not suitable;
and: - not suitable;
and: - not suitable;
and: - suitable. It remains only to remember that one of the roots is negative. Since their sum must be equal, then the root, which is smaller in absolute value, must be negative: . We check:
Answer:
Example #4:
Solve the equation.
Solution:
The equation is reduced, which means:
The free term is negative, and hence the product of the roots is negative. And this is possible only when one root of the equation is negative and the other is positive.
We select such pairs of numbers whose product is equal, and then determine which roots should have a negative sign:
Obviously, only roots and are suitable for the first condition:
Answer:
Example #5:
Solve the equation.
Solution:
The equation is reduced, which means:
The sum of the roots is negative, which means that at least one of the roots is negative. But since their product is positive, it means both roots are minus.
We select such pairs of numbers, the product of which is equal to:
Obviously, the roots are the numbers and.
Answer:
Agree, it is very convenient - to invent roots orally, instead of counting this nasty discriminant. Try to use Vieta's theorem as often as possible.
But the Vieta theorem is needed in order to facilitate and speed up finding the roots. To make it profitable for you to use it, you must bring the actions to automatism. And for this, solve five more examples. But don't cheat: you can't use the discriminant! Only Vieta's theorem:
Solutions for tasks for independent work:
Task 1. ((x)^(2))-8x+12=0
According to Vieta's theorem:
As usual, we start the selection with the product:
Not suitable because the amount;
: the amount is what you need.
Answer: ; .
Task 2.
And again, our favorite Vieta theorem: the sum should work out, but the product is equal.
But since it should be not, but, we change the signs of the roots: and (in total).
Answer: ; .
Task 3.
Hmm... Where is it?
It is necessary to transfer all the terms into one part:
The sum of the roots is equal to the product.
Yes, stop! The equation is not given. But Vieta's theorem is applicable only in the given equations. So first you need to bring the equation. If you can’t bring it up, drop this idea and solve it in another way (for example, through the discriminant). Let me remind you that to bring a quadratic equation means to make the leading coefficient equal to:
Excellent. Then the sum of the roots is equal, and the product.
It's easier to pick up here: after all - a prime number (sorry for the tautology).
Answer: ; .
Task 4.
The free term is negative. What's so special about it? And the fact that the roots will be of different signs. And now, during the selection, we check not the sum of the roots, but the difference between their modules: this difference is equal, but the product.
So, the roots are equal and, but one of them is with a minus. Vieta's theorem tells us that the sum of the roots is equal to the second coefficient with the opposite sign, that is. This means that the smaller root will have a minus: and, since.
Answer: ; .
Task 5.
What needs to be done first? That's right, give the equation:
Again: we select the factors of the number, and their difference should be equal to:
The roots are equal and, but one of them is minus. Which? Their sum must be equal, which means that with a minus there will be a larger root.
Answer: ; .
Let me summarize:
- Vieta's theorem is used only in the given quadratic equations.
- Using the Vieta theorem, you can find the roots by selection, orally.
- If the equation is not given or no suitable pair of factors of the free term was found, then there are no integer roots, and you need to solve it in another way (for example, through the discriminant).
3. Full square selection method
If all the terms containing the unknown are represented as terms from the formulas of abbreviated multiplication - the square of the sum or difference - then after the change of variables, the equation can be represented as an incomplete quadratic equation of the type.
For example:
Example 1:
Solve the equation: .
Solution:
Answer:
Example 2:
Solve the equation: .
Solution:
Answer:
In general, the transformation will look like this:
This implies: .
Doesn't it remind you of anything? It's the discriminant! That's exactly how the discriminant formula was obtained.
QUADRATIC EQUATIONS. BRIEFLY ABOUT THE MAIN
Quadratic equation is an equation of the form, where is the unknown, are the coefficients of the quadratic equation, is the free term.
Complete quadratic equation- an equation in which the coefficients are not equal to zero.
Reduced quadratic equation- an equation in which the coefficient, that is: .
Incomplete quadratic equation- an equation in which the coefficient and or free term c are equal to zero:
- if the coefficient, the equation has the form: ,
- if a free term, the equation has the form: ,
- if and, the equation has the form: .
1. Algorithm for solving incomplete quadratic equations
1.1. An incomplete quadratic equation of the form, where, :
1) Express the unknown: ,
2) Check the sign of the expression:
- if, then the equation has no solutions,
- if, then the equation has two roots.
1.2. An incomplete quadratic equation of the form, where, :
1) Let's take the common factor out of brackets: ,
2) The product is equal to zero if at least one of the factors is equal to zero. Therefore, the equation has two roots:
1.3. An incomplete quadratic equation of the form, where:
This equation always has only one root: .
2. Algorithm for solving complete quadratic equations of the form where
2.1. Solution using the discriminant
1) Let's bring the equation to the standard form: ,
2) Calculate the discriminant using the formula: , which indicates the number of roots of the equation:
3) Find the roots of the equation:
- if, then the equation has a root, which are found by the formula:
- if, then the equation has a root, which is found by the formula:
- if, then the equation has no roots.
2.2. Solution using Vieta's theorem
The sum of the roots of the reduced quadratic equation (an equation of the form, where) is equal, and the product of the roots is equal, i.e. , a.
2.3. Full square solution
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.
A task. 1. Two typists retyped the manuscript in 6 hours. 40 min. How long would it take for each typist to retype the manuscript, working alone, if the first one spent 3 hours more on this work than the second?
Solution. 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
The 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.
