Question

Find the value of k for which the equation x² − 2(1 + 3k) x + 7(3 + 2k) = 0 have equal roots.

Answer 1

Given:

x² − 2(1 + 3k) x + 7(3 + 2k) = 0

a = 1

b = −2(1 + 3k)

c = 7(3 + 2k)

For equal roots, the discriminant must be zero:

D = b² − 4ac = 0

Substitute values

[−2(1 + 3k)]² − 4(1) ⋅ 7(3 + 2k) = 0

4(1 + 3k)² − 28(3 + 2k) = 0

4(1 + 6k + 9k²) − 84 − 56k = 0

4 + 24k + 36k² − 84 − 56k = 0

36k² − 32k − 80 = 0

Divide throughout by 4:

9k² − 8k − 20 = 0

`k = (8 +- sqrt(64+720))/18`

`k = (8 +- sqrt784)/18`

`k = (8+-28)/18`

k = 2 or k = `-10/9`