Find the values of k for which the quadratic equation has equal roots. Also find the roots.

Given quadratic equation, (3k + 1) x² + 2(k + 1) x + 1 = 0

Comparing the given equation with ax² + bx + c = 0, we get

a = (3k + 1), b = 2(k + 1), c = 1

We know that a quadratic equation ax² + bx + c = 0 has equal roots (i.e., coincident roots), if b² – 4ac = 0.

∴ (2(k + 1))² – 4(3k + 1) (1) = 0

⇒ 4k² + 8k + 4 – 12k - 4 = 0

⇒ 4k² – 4k = 0

⇒ 4k(k – 1) = 0

By factorization method,

⇒ 7k² – 28k + 4k – 16 = 0

⇒ 7k (k – 4) + 4(k – 4) = 0

⇒ (k – 4) (7k + 4) = 0

⇒ 4k = 0 (or) (k - 1) = 0

⇒ k = 0 (or) k = 1

For equal roots, roots =

⇒ Roots =

Case 1: When k = 0

⇒ Roots = -(0 + 1)/2(3(0) + 1)

= -1/2

= -1/2, -1/2

Case 2: When k = 1

⇒ Roots = -(1 + 1)/2(3(1) + 1)

= -2/8

= -1/4, -1/4

The value of k is 0 or 1. The roots are -1/2, -1/2 or -1/4, -1/4.