For what values of k the expression will be a perfect square?

Question

the question is the expression $kx^2 +(k+1)x +2$ will be a perfect square of a linear polynomial for what values of k .

I am unable to understand the concept used in this question for finding the possible values for k.

please someone explain.

Answer 1

$$k\left(x^2+\dfrac{(k+1)x}{2k}+\left(\dfrac{k+1}{2k}\right)^2\right)+2-\dfrac{(k+1)^2}{4k}$$

So, we need $$2-\dfrac{(k+1)^2}{4k}=0\iff k=?$$

Answer 2

A quadratic has $2$ equal roots when its determinant is equal to $0$. So we have

$$(k+1)^2-8k=0$$ $$k^2-6k+1=0$$

at which point you can solve using your preferred method.