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

⇒ 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`