Question

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

Answer 1

Answer :

Given x = - 2 is root of equation 3x² + 7x + p = 0
So,
x = - 2 satisfied our given equation ( Gives equal to zero ) , So

$\Rightarrow$3 ( - 2 )² + 7 ( - 2 ) + p = 0

$\Rightarrow$3 ( 4 ) - 14 + p = 0

$\Rightarrow$12 - 14 + p = 0

$\Rightarrow$p = 2

And

Given equation x² + k ( 4x + k - 1 ) + p = 0 have equal roots ,
First we simplify our equation , and get

$\Rightarrow$x² + 4xk + k² - k + p = 0

As roots are equal , So we know

b² - 4ac = 0

Here b = 4k , a = 1 and c = k² - k + p , So

$\Rightarrow$( 4k )² - 4 ( 1 ) ( k² - k + p ) = 0

$\Rightarrow$16k² - 4 k² + 4k - 4 p = 0 , Substitute value of P , we get

$\Rightarrow$12k² + 4 k - 4 ( 2 ) - 0

$\Rightarrow$12 k² + 4k - 8 = 0

$\Rightarrow$3 k² + k - 2 = 0

$\Rightarrow$3 k² +3 k - 2k - 2 = 0

$\Rightarrow$3k ( k + 1 ) - 2 ( k + 1 ) = 0

$\Rightarrow$( 3k - 2 ) ( k + 1 ) = 0
So,

$\Rightarrow$k = - 1 and $\frac{2}{3}$