If -5 is a root of the quadratic equation 2x² + px - 15 = 0 and the quadratic equation p(x² + x) + k = 0 has equal roots, find the value of k.

As, -5 is the root of the quadratic equation 2x² + px - 15 = 0, so it will satisfy the equation.

∴ 2(-5)² + (-5)p - 15 = 0

⇒ 2 × 25 - 5p - 15 = 0

⇒ 50 - 5p - 15 = 0

⇒ 5p = 35

⇒ p = $\dfrac{35}{5}$ = 7.

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

⇒ 7(x² + x) + k = 0

⇒ 7x² + 7x + k = 0

Since, above equation has real and equal roots.

∴ D = 0

⇒ b² - 4ac = 0

⇒ (7)² - 4 × 7 × k = 0

⇒ 49 - 28k = 0

⇒ 28k = 49

⇒ k = $\dfrac{49}{28} = \dfrac{7}{4} = 1\dfrac{3}{4}$.

Hence, k = $1\dfrac{3}{4}$.