Question

Show that the equation x² + ax – 1 = 0 has real and distinct roots for all real values of x.

[Hint: D = (a² + 4) > 0.]

Answer 1

Given: Consider the quadratic equation x² + ax – 1 = 0

Step-wise calculation:

1. Compare with the standard form Ax² + Bx + C = 0: 

A = 1, B = a, C = –1

2. Compute the discriminant D = B² – 4AC

= a² – 4(1)(–1)

= a² + 4

3. For every real a, a² ≥ 0, so a² + 4 ≥ 4 > 0.

Hence, D = a² + 4 > 0 for all real a.

4. If the discriminant D > 0, a quadratic has two real and distinct roots.

Since D = a² + 4 > 0 for every real a, the equation x² + ax – 1 = 0 has two real and distinct roots for all real values of a.