# 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
*
< 0 , then
*
two
*
roots are
*
imaginary
*
numbers.

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