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

Given
One root of quadratic equation 2x² + px - 5 = 0 is 5.
2(5)² + p(5) - 5 = 0
⇒ 2 × 25 + 5p - 5 = 0
⇒ 50 + 5p - 5 = 0
⇒ 5p = -45
⇒ p = -9
Putting this value in quadratic equation p(x² + x) + k = 0
-9(x² + x) + k = 0
⇒ -9x² - 9x + k = 0
⇒ 9x² + 9x - k = 0
Comparing by ax² + bx + c = 0
a = 9, b = 9, c = -k
Roots of equation are equal
Discriminant (D) = 0
⇒ b² - 4ac = (9)² - 4 × 9 × (-k)
⇒ 0 = 81 + 36k
⇒ 36k = -81
⇒ k = -81/36
Thus, k = -9/4