Instruction

1

Write the quadratic equation in the form ax2 + bx + c = 0

Example:

Original equation: 12 + x2= 8x

Correct the equation: x2 - 8x + 12 = 0

Example:

Original equation: 12 + x2= 8x

Correct the equation: x2 - 8x + 12 = 0

2

Apply the vieta theorem, according to which the sum of the roots of the equation will be equal to the number "b", taken with the opposite sign, and their work - a "c".

Example:

In this equation b=-8, c=12, respectively:

x1+x2=8

x1∗x2=12

Example:

In this equation b=-8, c=12, respectively:

x1+x2=8

x1∗x2=12

3

Learn positive and negative numbers are roots of equations. If the product and the sum of the roots is positive number, each of the roots is positive. If the product of the roots is positive and the sum of the roots is negative, then both root – negative. If the product of the roots is negative, the roots one root is the " + " sign and another sign "-" In this case, you must use an additional rule: "If the sum of the roots is a positive number greater than the module root is also positive and if the sum of the roots is negative - the larger the module root is negative."

Example:

In this equation the sum and product of positive numbers: 8 and 12, then both the root - the positive number.

Example:

In this equation the sum and product of positive numbers: 8 and 12, then both the root - the positive number.

4

Solve this system of equations by choosing from the roots. Will be convenient to begin with the selection of the multipliers, and then, to check, substitute each pair of multipliers into the second equation and test whether the sum of these roots solution.

Example:

x1∗x2=12

Suitable pairs of roots are respectively: 12 and 1, 6 and 2, 4 and 3

Check out the pair using equation x1+x2=8. Pair

12 + 1 ≠ 8

6 + 2 = 8

4 + 3 ≠ 8

Accordingly, the roots of an equation are numbers 6 and 8.

Example:

x1∗x2=12

Suitable pairs of roots are respectively: 12 and 1, 6 and 2, 4 and 3

Check out the pair using equation x1+x2=8. Pair

12 + 1 ≠ 8

6 + 2 = 8

4 + 3 ≠ 8

Accordingly, the roots of an equation are numbers 6 and 8.

Note

In this example, was a variant of the quadratic equation where a=1. To the same way to solve a full quadratic equation, where a&ne 1, it is necessary to make an auxiliary equation, giving "a" to the unit.

Useful advice

Use this method for solving equations in order to quickly find the roots. Also it will help if you need to solve the equation in the mind, without resorting to the records.

# Advice 2 : What is the arithmetic square root

Any math operation has its opposite. Addition opposite of subtraction, multiplication and division. Have their "twins of opposites" and exponentiation.

Exponentiation implies that the given number must be multiplied by itself a certain number of times. For example, the construction of the number 2 in the fifth degree will be as follows:

2*2*2*2*2=64.

The number that should be multiplied by itself is called a basis of the degree and the number of multiplications is her figure. Exponentiation correspond to two opposite actions: finding rate and the finding of the base.

Finding the base of power is called the root. This means that you need to find the number that you want to build in a degree n to obtain this.

For example, you must remove the root of 4-th degree of the number 16, i.e., to determine what number to multiply on itself 4 times to get 16. This number – 2.

This arithmetic operation is written with a special sign of the radical: √, which is indicated on the left of the exponent.

If the exponent is an even number, the root may be two numbers with the same module, but with different signs – positive and negative. So, in the example it can be numbers 2 and -2.

The expression must be unambiguous, i.e. it must have one result. For this and introduced the concept of arithmetic root, which may be only positive number. To be less than zero, the arithmetic root can not.

Thus, in the above example, the arithmetic root is just the number 2, and the second answer is -2 is excluded by definition.

For some degrees, which are used most frequently in mathematics, there are special titles that are originally associated with the geometry. We are talking about the construction of the second and third degree.

Second degree erecting the side length of a square when you need to calculate its area. If you need to find the volume of a cube, the length of its edges raised to the third degree. Therefore, the second degree is called a square number, and the third cube.

Accordingly, the root of the second degree is called square, and the root of the third degree cubic. Square root – the only one of the roots, in which over a radical is the exponent:

√64=8

So, the arithmetic square root of a given number is a positive number, which is necessary to build in the second degree, to obtain the number.

2*2*2*2*2=64.

The number that should be multiplied by itself is called a basis of the degree and the number of multiplications is her figure. Exponentiation correspond to two opposite actions: finding rate and the finding of the base.

## The root

Finding the base of power is called the root. This means that you need to find the number that you want to build in a degree n to obtain this.

For example, you must remove the root of 4-th degree of the number 16, i.e., to determine what number to multiply on itself 4 times to get 16. This number – 2.

This arithmetic operation is written with a special sign of the radical: √, which is indicated on the left of the exponent.

## Arithmetic root

If the exponent is an even number, the root may be two numbers with the same module, but with different signs – positive and negative. So, in the example it can be numbers 2 and -2.

The expression must be unambiguous, i.e. it must have one result. For this and introduced the concept of arithmetic root, which may be only positive number. To be less than zero, the arithmetic root can not.

Thus, in the above example, the arithmetic root is just the number 2, and the second answer is -2 is excluded by definition.

## Square root

For some degrees, which are used most frequently in mathematics, there are special titles that are originally associated with the geometry. We are talking about the construction of the second and third degree.

Second degree erecting the side length of a square when you need to calculate its area. If you need to find the volume of a cube, the length of its edges raised to the third degree. Therefore, the second degree is called a square number, and the third cube.

Accordingly, the root of the second degree is called square, and the root of the third degree cubic. Square root – the only one of the roots, in which over a radical is the exponent:

√64=8

So, the arithmetic square root of a given number is a positive number, which is necessary to build in the second degree, to obtain the number.