Question

Find the values of k for which the following equation has equal roots:

(k + 1)x² + 2(k + 3)x + (k + 8) = 0

Answer 1

Given: (k + 1)x² + 2(k + 3)x + (k + 8) = 0

Step-wise calculation:

1. For a quadratic ax² + bx + c = 0 equal (repeated) roots occur when discriminant D = b² – 4ac = 0.

2. Here a = k + 1, b = 2(k + 3), c = k + 8.

3. D = [2(k + 3)]² – 4(k + 1)(k + 8)

= 4[(k + 3)² – (k + 1)(k + 8)] 

= 4[(k² + 6k + 9) – (k² + 9k + 8)] 

= 4(–3k + 1)

4. Set D = 0

⇒ 4(–3k + 1) = 0

⇒ –3k + 1 = 0 

⇒ `k = 1/3`

5. Also require a ≠ 0 so the equation is quadratic: k ≠ –1.

`k = 1/3`