A Discriminant is a parameter of an object or system calculated as an aid to its classification or solution.
For a quadratic equation ax2 + bx + c = 0, the discriminant is b2 − 4ac.
The roots of a quadratic equation with real coefficients:
- are real and distinct if the discriminant is positive
- are real with at least two equal if the discriminant is zero
- and include a conjugate pair of complex roots if the discriminant is negative.
This is a quadratic in x where:
Show the roots of x2 – (a-1)x – 2(a-1)2 = 0 are real
a =1, b = -(a-1) = 1- a, and c = – 2(a-1)2 = -2a2+2a-1.
So b2 − 4ac = (12+a2-2a) – 4(1)(-2a2+2a-1)
= 9a2 + 6 – 3
Since 9a2 must be either 0 or positive and + 6 – 3 is positive, the discriminant is positive.
Therefore the roots are real.