Properties of roots of a quadratic equation. 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 x²+ px + q = 0 is equal to coefficient at the first power of unknown, taken with a back sign, i.e.                                                    

 x₁ +  x₂ = – p ,

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

x₁  ·  x₂  =  q .

To prove Viete’s theorem, use the formula, by which roots of reduced quadratic equation are calculated.

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