Activity

Factoring Perfect Square Trinomials

To factor perfect square trinomials, follow the formulas:

$a^2+2ab+b^2=(a+b{)}^2$

$a^2-2ab+b^2=(a-b{)}^2$

Let's look at an example of how to factor perfect square trinomials.

Example

Factor $9x^2+12x+4$.

Step 1 - Does the trinomial fit the perfect square trinomial pattern, $a^2+2ab+b^2$? If so, write the first term as $a^2$ and last constant as $b^2$.

• Is the first term a perfect square? Write it as a square, $a^2$. Is $9x^2$ a perfect square? Yes — write it as $(3x{)}^2$. $\begin{array}{c}9x^2+12x+4\hfill \\ (3x{)}^2\hfill \end{array}$

• Is the last term a perfect square? Write it as a square, $b^2$. Is 4 a perfect square? Yes — write it as $(2{)}^2$. $\begin{array}{c}9x^2+12x+4\hfill \\ (3x{)}^2(2{)}^2\hfill \end{array}$

• Check the middle term. Is it $2ab$? Is $12x$ twice the product of $3x$ and 2? Does it match? Yes, so we have a perfect trinomial.

$\begin{array}{c}\hfill 9x^2+12x+4\hfill \\ \hfill (3x{)}^2(2{)}^2\hfill \\ \hfill (3x{)}^2(2{)}^2\hfill \\ \hfill \searrow \swarrow \hfill \\ \hfill 2(3x)(2)\hfill \\ \hfill 12x\hfill \end{array}$

Step 2 - Factor the binomial into: $a^2+2ab+b^2=(a+b{)}^2$.

$9x^2+12x+4$

$\begin{array}{c}{a}^2+2·a·b+b^2\hfill \\ (3x{)}^2+2·3x·2+2^2\hfill \\ (a+b{)}^2\hfill \\ (3x+2{)}^2\hfill \end{array}$

Step 3 - Check.

$(3x+2{)}^2$

$(3x{)}^2+2·3x·2+2^2$

$9x^2+12x+4✔$