Quadratic equation

Question

Here's the question below -

$x^2 - 6x + (p^2 - 6 )^2$ is a perfect square , write down the possible values of p .

My thoughts :

I thought of this expansion method -

$(A-B)^2 = A^2 - 2 AB + B^2$

But I'm not sure how to start ..

Answer 1

Hint

There is no need to complicate matters as the path you have indicated. See if the following looks simple and intuitive enough

The only way to complete the square for $x^2 - 6x$ is by adding $9$ to it. So $(p^2 - 6)^2 = 9$. Can you now find the values of $p$ from here ?

Answer 2

You are right. $A=x$ $B=3$ and so, $(P^2-6)^2=9$ which means $P^2-6=+/- 3$ and so, $P^2=9$ or $3$

$P=3$ , $-3$ , $\sqrt{3}$ , $-\sqrt{3}$