Question

Discuss the nature of the roots of the following equation without actually solving it:

x² – 8x + 7 = 0

Answer 1

Given: x² – 8x + 7 = 0 so a = 1, b = –8, c = 7

Step-wise calculation:

1. Discriminant D = b² – 4ac 

= (–8)² – 4 × 1 × 7 

= 64 – 28

= 36

2. Since D > 0, the roots are real and unequal.

The discriminant test: D > 0 ⇒ Two distinct real roots.

3. The quadratic formula gives roots = `(-b ± sqrt(D))/(2a)`.

Here `sqrt(D) = sqrt(36) = 6`, an integer, so `(-b ± sqrt(D))/(2a) = (8 ± 6)/2` are rational numbers a ratio of integers.

Therefore, the roots are rational.

Because D = 36 > 0, the roots are real and unequal; because D is a perfect square (36) and a, b, c are integers, `sqrt(D)` is an integer, so the two roots are rational.