Question

Find the value of ‘k’ for which the equation 5x² − 4x + 2 + k (4x² − 2x − 1) = 0 has real and equal roots.

Answer 1

Given:

5x² − 4x + 2 + k(4x² − 2x − 1) = 0

5x² − 4x + 2 + 4kx² − 2kx − k = 0

(5 + 4k) x² + (−4 − 2k) x + (2 − k) = 0

For real and equal roots, the discriminant = 0.

a = 5 + 4k, b = −4 − 2k, c = 2 − k

b² − 4ac = 0

(−4 − 2k)² − 4(5 + 4k) (2 − k) = 0

(4 + 2k)² − 4[(5 + 4k) (2 − k)] = 0

(16 + 16k + 4k²) − 4(10 + 3k − 4k²) = 0

16 + 16k + 4k² − 40 − 12k + 16k² = 0

20k² + 4k − 24 = 0

Divide throughout by 4:

5k² + k − 6 = 0

(5k + 6) (k − 1) = 0

` k = -6/5 or k = 1`