If -2 is a root of the equation 3x² + 7x + p = 1, find the value of p. Now find the value of k so that the roots of the equation x² + k(4x + k - 1) + p = 0 are equal.

Since, -2 is a root of the equation 3x² + 7x + p = 1,

∴ 3(-2)² + 7(-2) + p = 1

⇒ 3(4) - 14 + p = 1

⇒ 12 - 14 + p = 1

⇒ -2 + p = 1

⇒ p = 3.

Substituting value of p in x² + k(4x + k - 1) + p = 0 we get,

⇒ x² + k(4x + k - 1) + 3 = 0

⇒ x² + 4kx + k² - k + 3 = 0

Since, roots are equal, D = 0.

∴ b² - 4ac = 0

⇒ (4k)² - 4(1)(k² - k + 3) = 0

⇒ 16k² - 4k² + 4k - 12 = 0

⇒ 12k² + 4k - 12 = 0

⇒ 4(3k² + k - 3) = 0

⇒ 3k² + k - 3 = 0

$\Rightarrow k = \dfrac{-1 \pm \sqrt{(1)^2 - 4(3)(-3)}}{2(3)} \\[1em] = \dfrac{-1 \pm \sqrt{1 + 36}}{6} \\[1em] = \dfrac{-1 \pm \sqrt{37}}{6}.$

Hence, p = 3 and k = $\dfrac{-1 \pm \sqrt{37}}{6}$.