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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.
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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
⇒ p = 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) + 3 = 0
x2 + 4kx + (k2 − k + 3) = 0
This equation will have equal roots, if the discriminant is zero.
Here,
a = 1
b = 4k
c = k2 − k + 3
∴ Discriminant, D = (4k)2 − 4(k2 − k + 3) = 0
⇒ 16k2 − 4k2 + 4k − 12 = 0
⇒ 12k2 + 4k − 12 = 0
⇒ 3k2 + k − 3 = 0
On comparing with ax2 + bx + c = 0
We have a = 3, b = 1, c = −3
Then by quadratic formula, we have
x = `(-b +- sqrt(b^2 - 4ac))/(2 a)`
x = `(-1 +- sqrt(1^2 - 4 xx 3 xx (-3)))/(2 xx 3)`
x = `(-1 +- sqrt(1 + 36))/(2 xx 3)`
x = `(-1 +- sqrt(1 + 36))/6`
i.e. `(-1 +- sqrt37)/6`
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