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

9.4.2 Rewrite Squared Expressions

9.4.2 • Rewrite Squared Expressions

Activity

1. Write each expression in standard form:

a. $(4 x + 1)^{2}$

Compare your answer:

$16 x^{2} + 8 x + 1$

b. $(5 x - 2)^{2}$

Compare your answer:

$25 x^{2} - 20 x + 4$

c. $(\frac{1}{2} x + 7)^{2}$

Compare your answer:

$\frac{1}{4} x^{2} + 7 x + 49$

d. $(3 x + n)^{2}$

Compare your answer:

$9 x^{2} + 2 (3) (n) x + n^{2}$ or $9 x^{2} + 6 n x + n^{2}$

e. $(k x + m)^{2}$

Compare your answer:

$k^{2} x^{2} + 2 k m x + m^{2}$

2. Decide if each expression is a perfect square. If so, write an equivalent expression of the form $(k x + m)^{2}$ . If not, suggest one change to turn it into a perfect square.

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

Compare your answer:

Yes, a perfect square. $(2 x + 3)^{2}$

g. $4 x^{2} + 8 x + 25$

Compare your answer:

No, not a perfect square. Suggested changes include:

Changing the coefficient of the linear term to 20 gives $4 x^{2} + 20 x + 25$ , which is equivalent to $(2 x + 5)^{2}$ .
Changing the coefficient of the linear term to -20 gives $4 x^{2} - 20 x + 25$ , which is equivalent to $(2 x - 5)^{2}$ .
Changing the constant term to 4 gives $4 x^{2} + 8 x + 4$ , which is equivalent to $(2 x + 2)^{2}$ .

Video: Rewrite Squared Expressions

Watch the following video to learn more about rewriting squared expressions.

Rewrite Squared Expressions

Self Check

Find the equivalent factored expression for

4x^2+12x+9

.

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

Factoring Perfect Square Trinomials

To factor perfect square trinomials, follow the formulas:

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

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

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

Example

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

Step 1 - Does the trinomial fit the perfect square trinomial pattern, $a^{2} + 2 a b + b^{2}$ ? 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 $9 x^{2}$ a perfect square? Yes — write it as $(3 x)^{2}$ . $\begin{matrix} 9 x^{2} + 12 x + 4 \\ (3 x)^{2} \end{matrix}$

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

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

$\begin{matrix} 9 x^{2} + 12 x + 4 \\ (3 x)^{2} (2)^{2} \\ (3 x)^{2} (2)^{2} \\ ↘ ↙ \\ 2 (3 x) (2) \\ 12 x \end{matrix}$

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

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

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

Step 3 - Check.

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

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

$9 x^{2} + 12 x + 4 ✔$

Try it

Factoring Perfect Square Trinomials

Factor $9 x^{2} - 30 x + 25$ .

Here is how to factor a perfect square trinomial:

Step 1 - Write the first term as $a^{2}$ and the last constant as $b^{2}$ .

$a^{2} = (3 x)^{2}$ , $b^{2} = 5^{2}$

Make sure the middle term is $2 a b$ .
$2 a b = 2 (3 x) (5) = 30 x$

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

$(3 x - 5)^{2}$

Step 3 - Check.

$(3 x - 5) (3 x - 5)$

$= 9 x^{2} - 15 x - 15 x + 25$

$= 9 x^{2} - 30 x + 25$