# What is the difference between perfect square and difference of squares?

Jul 7, 2015

A perfect square can be factored as ${\left(p + q\right)}^{2}$
The difference of squares can be factored as $\left(p + q\right) \left(p - q\right)$

#### Explanation:

I assume here that we are dealing with polynomials that can be factored into binomials.

For trinomials of the form
$\textcolor{w h i t e}{\text{XXXX}}$$a {x}^{2} + b x + c$

$a {x}^{2} + b x + c$ is a perfect square if
$\textcolor{w h i t e}{\text{XXXX}}$${\exists}_{p} | {p}^{2} = a {x}^{2}$
$\textcolor{w h i t e}{\text{XXXX}}$${\exists}_{q} | {q}^{2} = c$
and
$\textcolor{w h i t e}{\text{XXXX}}$$p \cdot q = b x$

Since the difference of squares factors as
$\textcolor{w h i t e}{\text{XXXX}}$$\left(p + q\right) \left(p - q\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$= {p}^{2} - {q}^{2}$
an obvious requirement for $a {x}^{2} + b x + c$ to be the difference of squares is that $b = 0$