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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.
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Solution
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)`.
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