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.