# How do you know if x^2 - 24x + 144 is a perfect square trinomial and how do you factor it?

Jun 9, 2015

Comparing with the form ${a}^{2} - 2 a b + {b}^{2} = {\left(a - b\right)}^{2}$, we find ${x}^{2} - 24 x + 144 = {\left(x - 12\right)}^{2}$

#### Explanation:

All perfect square trinomials are of the form:

${a}^{2} \pm 2 a b + {b}^{2} = {\left(a \pm b\right)}^{2}$

In our case we see:

${x}^{2} - 24 x + 144$

$= {x}^{2} - 24 x + {12}^{2}$

$= {x}^{2} - \left(2 \cdot x \cdot 12\right) + {12}^{2}$

which is of the form ${a}^{2} - 2 a b + {b}^{2}$ with $a = x$ and $b = 12$, so

$= {\left(x - 12\right)}^{2}$