for every positive integer $k$, there exists an integer $x$ such that $kx^2-1$ is quadratic residue (mod $p$)

Question

Prove that for every positive integer $k$, there exists an integer $x$ such that $kx^2-1$ is quadratic residue (mod $p$)

I don't think this statement is true since $k$ the sequence $k \cdot 1^2-1,k \cdot 2^2-1 , \cdots , k \cdot \left( \frac{p-1}{2} \right)^2-1$ are congruent to $r_1,r_2, \cdots , r_{(p-1)/2}$ modulo $p$ where as $r_i$ are quadratic residue (mod $p$) for some $k$.

My question: Is the statement above true ? If not then for what $k$ it's true ?