Question

For what value of p the quadratic equation $p{x}^2-18x+1=0$ is a perfect square?

Answer 1

For perfect square,
$b^2-4ac=0$
a = p b = -18 c = 1
$b^2-4ac=(-18{)}^2-4\times p\times 1=0$
$324=4\times p$
$P=\frac{324}{4}=81$

If it is a perfect square, the quadratic equation will have equal roots.

Example
$(x+2{)}^2=0$
$x+2=0$
$x=-2$
Then according to quadratic formula, Discriminant should be equal to zero.