# Properties of roots of a quadratic equation. Viete’s theorem

*Roots of quadratic equation. Discriminant. Viete’s theorem.*

The formula

shows, that the three cases are possible:

1) * b *^{2}* *– * 4 * * a
c* > 0 , then

*two*

roots are

*different real*numbers;

2) * b*^{ 2}* *– * 4 * * a
c* = 0 , then

*two*roots

are

*equal real*numbers;

3) * b *^{2}* *– * 4 * * a
c* two

roots are

*imaginary*numbers.

The expression * b *^{2}* *– * 4** a c *,*
*value of which permits to differ these three cases,

is called a

**of**

*discriminant*a quadratic equation and marked as

*D*.

* *

*Viete’s theorem.* *A sum of roots of reduced quadratic equation*

*x*

^{2}

*+ px + q*= 0

*is equal to coefficient at the first power of unknown, taken with a back sign, i.e.*

* *

* x*_{1}* + x*_{2}* * =* * –* **p ,*

*and a product of the roots is equal
to a free term, i.e.*

* *

*x*_{1}* * ·*
x*

_{2}

*=*

*q .*

To prove Viete’s theorem,

use the formula, by which roots of *reduced* quadratic equation are calculated.