Question

If x= -2 is a root of equation 3x ² + 7x + p = 0, then find the value of k so that the roots of the equation x² + k(4x + k - 1) + p=0 are equal.

Answer 1

3x ² + 7x + p = 0

Put x = -2
We get

3(-2) ² + 7(-2) + p = 0

3*4 - 14 + p = 0

12 - 14 + p = 0

p = 2
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x² + k(4x + k - 1) + p = 0

Put p = 2
We get

x² + k(4x + k - 1) + 2 = 0

=> x² + 4kx + k² - k + 2 = 0

Comparing this equation with ax² + bx + c = 0
We have
a = 1
b = 4k
c = k² - k + 2

Given x² + k(4x + k - 1) + p = 0 has equal roots.
So,
b² - 4ac = 0
or,
b² = 4ac

On putting values of a, b, c we get

(4k)² = 4(1)(k² - k + 2)

=>
16k² = 4k² -4k + 8

=>
12k² + 4k - 8 = 0

=>
3k² + k - 2 = 0

=>
(k + 1)(3k - 2) = 0

=>
k = -1 or 2/3