Problem Solving (PS) | Re: If k is a negative constant and x is a variable, what is the value of

Asked: If k is a negative constant and x is a variable, what is the value of k if one of the roots of the equation k(x - 1)^2 = 5x - 7 is double theother?


\( k(x - 1)^2 = 5x -7\)
\( k(x^2 - 2x + 1) = kx^2 - 2kx + k = 5x-7\)
\( kx^2 - (2k+5) + k+7 =0\)

Let the roots of the equation be r and2r

\( r + 2r = 3r = (2k + 5)/k;\)
\( r =(2k+5)/3k\) (1)
\( 2r^2 = (k+7)/k\)
\( r^2 =(k+7)/2k\) (2)

\( (k+ 7)/2k =(2k+5)^2/9k^2\)
\( 9k^2(k+7) = 2k(2k+5)^2 = 2k(4k^2 + 20k + 25) =8k^3+40k+50\)
\( 9k^3 + 63k^2 =8k^3+40k+50k\)
\( k^3 + 23k^2 - 50k =0\)
k(k+25)(k-2)=0

k = -25; since k is a negative integer

IMOE