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