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Question
If x = −2 is a root of the equation 3x2 + 7x + p = 1, find the values of p. Now find the value of k so that the roots of the equation x2 + k(4x + k − 1) + p = 0 are equal.
Solution
Since −2 is a root of the equation 3x2 + 7x + p = 1,
3(−2)2 + 7(−2) + p = 1
`=>` 12 − 14 + p = 1
`=>` −2 + p = 1
`=>` p = 1 + 2 = 3
∴ The equation becomes 3x2 + 7x + p = 1.
Putting p = 3 in x2 + k(4x + k − 1) + p = 0, we get
x2 + k(4x + k − 1) + 2 = 0
x2 + 4kx + (k2 − k + 2) = 0
This equation will have equal roots, if the discriminant is zero.
Here,
a = 1
b = 4k
c = k2 − k + 2
∴ Discriminant, D = (4k)2 − 4(k2 − k + 2) = 0
`=>` 16k2 − 4k2 + 4k − 8 = 0
`=>` 12k2 + 4k − 8 = 0
`=>` 3k2 + k − 2 = 0
`=>` 3k2 + 3k − 2k − 2 = 0
`=>` 3k(k + 1) − 2(k + 1) = 0
`=>` (3k − 2)(k + 1) = 0
`=>` 3k − 2 = 0 or k + 1 = 0
`=>` k = `2/3` or k = −1
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