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.

Answer 1

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