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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.
Solution
-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)`.
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