Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

Question

Find the roots of each of the following equations, if they exist, by applying the quadratic formula: x 2 + 5x – (a 2 + a – 6) = 0

Philoid · Accepted Answer

Given: x² + 5x – (a² + a – 6) = 0

Comparing with standard quadratic equation Ax² + Bx + C = 0

A = 1, B = 5, C = – (a² + a – 6)

Discriminant D = B² – 4AC

= (5)² – 4.1. – (a² + a – 6)

= 25+ 4a² + 4a – 24 = 4a² + 4a + 1

= (2a + 1)² > 0 Using a² + 2ab + b² = (a + b)²

Hence the roots of equation are real.

= (2a + 1)

Roots are given by

x = (a – 2) or x = – (a + 3)

Hence the roots of equation are (a – 2) or x = – (a + 3)