Question

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.

Answer 1

-5 is a root of the quadratic equation
2x² + px – 15 = 0, then
⇒ 2(5)² – p( -5) – 15 = 0
⇒ 50 – 5p – 15 = 0
⇒ 35 – 5p = 0
⇒  5p = 35 = 0

⇒ p = `(35)/(5)` = 7
p(x² + x) + k = 0 has equal roots
⇒  px² + px + k = 0
⇒  7x² + 7x + k = 0
Here, a = 7, b = 7, c = k
b² - 4ac
= (7)² - 4 x 7 x k
= 49 - 28k
∵ Roots are equal
∴ b² - 4ac = 0
⇒ 49 - 28k = 0
⇒ 28k = 49

⇒ k = `(49)/(28) = (7)/(4)`

∴ k = `(7)/(4)`.