Question

Find the values of k for which each of the following quadratic equation has equal roots: 9x² + kx + 1 = 0 Also, find the roots for those values of k in each case.

Answer 1

9x² + kx + 1 = 0
Here a = 9, b = k, c = 1
∴ D = b² - 4ac
= k² - 4 x 9 x 1
= k² - 36
∵ Roots are equal.
∴ D = 0
⇒ k² - 36 = 0
⇒ (k + 6)(k - 6) = 0
EIther k + 6 = 0, then k = -6
k - 6 = 0, then k = 6
∴  k = 6, -6
(a) If k = 6, then
9x² + 6x + 1 = 0
⇒ (3x)² + 2 x 3x x 1 + (1)² = 0
⇒ (3x + 1)² = 0
∴ 3x + 1 = 0
⇒ 3x = -1

x = `-(1)/(3),(1)/(3)`
(b) If k = -6, then
9x² - 6x + 1 = 0
⇒ (3x)² - 2 x 3x  x 1 + (1)² = 0
⇒ (3x - 1)² = 0
⇒ 3x - 1 = 0
⇒ 3x = 1

⇒ x = `(1)/(3)`

x = `(1)/(3),(1)/(3)`.