Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Mini Lesson Question

Question #2: Factoring Perfect Square Trinomials

Which of the following expressions is equivalent to

4x^2-12x+9

?

$4(x^2-3x+9)$
$(2x-3)(2x+3)$
$(2x+3)^2$
$(2x-3)^2$

Factor Perfect Square Trinomials

Some trinomials are perfect squares. They result from multiplying a binomial times itself.

${(\begin{matrix} a + b \\ 3 x + 4 \end{matrix})}^{2}$

$a^{2} + 2 \cdot a \cdot b + b^{2} {(3 x)}^{2} + 2 (3 x \cdot 4) + 4^{2}$

$9 x^{2} + 24 x + 16$

The trinomial $9 x^{2} + 24 x + 16$ is called a perfect square trinomial. It is the square of the binomial $3 x + 4$ .

If you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomial pattern, you will save yourself a lot of work.

Here is the pattern: the reverse of the binomial squares pattern.

Perfect Square Trinomials Pattern

If $a$ and $b$ are real numbers, then
$a^{2} + 2 a b + b^{2} = (a + b)^{2}$
$a^{2} - 2 a b + b^{2} = (a - b)^{2}$

Example

Here is an example of how to factor a perfect square trinomial.

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

Step 1 - Does the trinomial fit the perfect square trinomials pattern, $a^{2} + 2 a b + b^{2}$ ?

Is the first term a perfect square? Write it as a square, $a^{2}$ .
Is the last term a perfect square? Write it as a square, $b^{2}$ .
Check the middle term. Is it $2 a b$ ?

Is $9 x^{2}$ a perfect square? Yes: write it as $(3 x)^{2}$ . Is 4 a perfect square? Yes: write it as $(2)^{2}$ . Is $12 x$ twice the product of $3 x$ and 2? Does it match? Yes, so we have a perfect square trinomial.

$9 x^{2} + 12 x + 4 (3 x)^{2} (3 x)^{2} (2)^{2} (3 x)^{2} (2)^{2} ↘ ↙ 2 (3 x) (2) 12 x$

Step 2 - Write the square of the binomial.

$9 x^{2} + 12 x + 4$

$a^{2} + 2 \cdot a \cdot b + b^{2} (3 x)^{2} + 2 \cdot 3 x \cdot 2 + 2^{2}$

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

Step 3 - Check.

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

$9 x^{2} + 12 x + 4 ✓$

Try it

Try It: Factor Perfect Square Trinomials

Factor $81 y^{2} - 72 y + 16$ .

Here is how to factor the perfect square trinomial:

$81 y^{2} - 72 y + 16$

Step 1 - Are the first and last terms perfect squares?

$(9 y)^{2} 4^{2}$

Is the middle term $2 a b$ ?

$2 (9 y) (4) = 72 y$

Step 2 - Write the square of the binomial.

$(9 y - 4)^{2}$

Step 3 - Check.

$(9 y - 4)^{2}$

$(9 y - 4) \cdot (9 y - 4)$

$(9 y)^{2} - 36 y - 36 y + 4^{2}$

$81 y^{2} - 72 y + 16 ✓$

Check Your Understanding

Which of the following is equivalent to $36 y^{2} + 12 y + 1$ ?

Multiple Choice:

$(6 y + 1) (6 y - 1)$
$(6 y + 1)^{2}$
$(6 y - 1)^{2}$
$12 (3 y^{2} + y + 1)$

Check yourself: $(6 y + 1)^{2} = (6 y + 1) (6 y + 1)$ , then, using the distributive property, becomes $36 y^{2} + 6 y + 6 y + 1$ , which becomes the given perfect square trinomial.

Try again. Take a moment to think about what you learned in the mini-lesson review.

Videos: Factoring Perfect Squares and Identifying Perfect Square Form

Watch the following videos to learn more about factoring perfect square trinomials and perfect square form.

Khan Academy: Factoring Perfect Squares

Access multimedia content

Khan Academy: Identifying Perfect Square Form