Question

For what values of k, does the quadratic equation: kx2-(k-1)x +(k-1)=0 becomes a perfect square?

Answer 1

you mean to say equation Kx² -(k - 1)x + (k - 1) = 0 becomes perfect square . it is possible only when the roots of this quadratic equation will be same.

Let a is a root of equation then other root must be a
sum of roots = a + a = (k - 1)/k
2a = (k -1)/k
a = (k - 1)/2k -------(1)

product of roots = a.a = a² = (k -1)/k
a = ±$\bold{\sqrt{\frac{k-1}{k}}}$ ----(2)

now from equations (1) and (2)
$\bold{\frac{k-1}{2k}}=\pm\bold{\sqrt{\frac{k-1}{k}}}$
$\bold{\sqrt{\frac{k-1}{4k}}}.{\bold{\sqrt{\frac{k-1}{k}}}}=\pm\bold{\sqrt{\frac{k-1}{k}}}\\\\\bold{\sqrt{\frac{k-1}{4k}}}=\pm1$
taking square both sides,
(k - 1)/4k = 1
k - 1 = 4k
k = -1/3

Hence, k = -1/3