# How do you know if  9a^2 − 30a + 25 is a perfect square trinomial and how do you factor it?

Jun 27, 2015

Notice that $9 {a}^{2} = {\left(3 a\right)}^{2}$ and $25 = {5}^{2}$

Need to check that the middle term is $\pm 2 \cdot 3 a \cdot 5 = \pm 30 a$ - Yes.

$9 {a}^{2} - 30 a + 25 = {\left(3 a - 5\right)}^{2}$

#### 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 example, we can recognise $A = 3 a$ and $B = 5$, so just need to check that the middle term is correct.

$2 A B = 2 \times 3 a \times 5 = 30 a$

So we have ${A}^{2} - 2 A B + {B}^{2} = {\left(A - B\right)}^{2} = {\left(3 a - 5\right)}^{2}$