Find the value of k for which the following equation has equal roots.

1.	Find the value of k for which the following equation has equal roots. x2 + 4kx + (k2 – k + 2) =0
Answer» Given quadratic equation is : x² + 4kx + (k² – k + 2) = 0 For equal roots, we have b² – 4 ac = 0 ⇒ (4k)² – 4(1) (k²- k + 2) = 0 ⇒ 16k² – 4k² + 4k – 8 = 0 ⇒ 12k² + 4k – 8 = 0 or 3k² + k – 2 = 0 ⇒ 3k² + 3k – 2k - 2 = 0 ⇒ 3k(k + 1) – 2(k + 1) = 0 ⇒ (k + 1)(3k – 2) = 0 ⇒ k + 1 = 0 or 3k – 2 = 0 k = – 1 or k = 2/3

Find the value of k for which the following equation has equal roots. x2 + 4kx + (k2 – k + 2) =0

Answer»

Given quadratic equation is :

x² + 4kx + (k² – k + 2) = 0

For equal roots, we have

b² – 4 ac = 0

⇒ (4k)² – 4(1) (k²- k + 2) = 0

⇒ 16k² – 4k² + 4k – 8 = 0

⇒ 12k² + 4k – 8 = 0

or 3k² + k – 2 = 0

⇒ 3k² + 3k – 2k - 2 = 0

⇒ 3k(k + 1) – 2(k + 1) = 0

⇒ (k + 1)(3k – 2) = 0

⇒ k + 1 = 0 or 3k – 2 = 0

k = – 1 or k = 2/3