# How do you know if 4x^2 - 20x +25 is a perfect square trinomial and how do you factor it?

Jun 6, 2015

Perfect square trinomials are of the form ${a}^{2} \pm 2 a b + {b}^{2} = {\left(a \pm b\right)}^{2}$.

In the case of $4 {x}^{2} - 20 x + 25$, we can look at the first and last terms to notice:

$4 {x}^{2} = {\left(2 x\right)}^{2}$
$25 = {5}^{2}$

...so if it is a square trinomial, we must have $a = 2 x$ and $b = 5$.

Then $2 a b = 2 \cdot 2 x \cdot 5 = 20 x$ matching our middle term with a minus sign.

So:

$4 {x}^{2} - 20 x + 25 = {\left(2 x\right)}^{2} - \left(2 \cdot \left(2 x\right) \cdot 5\right) + {5}^{2} = {\left(2 x - 5\right)}^{2}$