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.
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