Question

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

x² + 4kx + (k² – k + 2) = 0

Answer 1

Given: Equation: x² + 4kx + (k² – k + 2) = 0

Step-wise calculation:

1. For equal roots the discriminant must be zero: D = b² – 4ac.

2. Here a = 1, b = 4k, c = k² – k + 2

So, D = (4k)² – 4 × 1 × (k² – k + 2)

= 16k² – 4k² + 4k – 8

= 12k² + 4k – 8

3. Set D = 0: 12k² + 4k – 8 = 0

Divide by 4: 3k² + k – 2 = 0

4. Factor the quadratic:

(3k – 2)(k + 1) = 0 

⇒ k = `2/3` or k = –1

The equation has equal roots for k = `2/3` or k = –1.