Question

Find the values of k so that the quadratic equation (4 – k) x² + 2 (k + 2) x + (8k + 1) = 0 has equal roots.

Answer 1

(4 – k) x² + 2 (k + 2) x + (8k + 1) = 0
Here a = (4 – k), b = 2 (k + 2), c = 8k + 1
∴ D = b² – 4ac
= [2(k + 2)]² – 4 x (4 – k)(8k + 1) = 0
= 4(k + 2)² - 4(32k + 4 – 8k² – k)
= 4(k² + 4k + 4) –4(32k + 4 – 8k² – k)
= 4k² + 16k + 16 - 128k – 16 + 32k² + 4k
= 36k² – 108k
= 36k(k – 3)
∵ Roots are equal
∴ D = 0
⇒ 36k(k – 3) = 0
⇒ k(k – 3) = 0
Either k = 0
or
k – 3 = 0,
then k= 3
k = 0, 3.