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

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Algebra 1

9.1.2 Recognizing Structure in Perfect-Square Expressions

Algebra 19.1.2 Recognizing Structure in Perfect-Square Expressions

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Activity

1. Each expression is written as the product of factors. Write an equivalent expression in standard form.

a. $(3 x)^{2}$

Compare your answer:

$9 x^{2}$

b. $7 x \cdot 7 x$

Compare your answer:

$49 x^{2}$

c. $(x + 4) (x + 4)$

Compare your answer:

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

d. $(x + 1)^{2}$

Compare your answer:

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

e. $(x - 7)^{2}$

Compare your answer:

$x^{2} - 14 x + 49$

f. $(x + n)^{2}$

Compare your answer:

$x^{2} + 2 n x + n^{2}$

2. Why do you think the following expressions can be described as perfect squares?

$x^{2} + 6 x + 9 x^{2} - 16 x + 64 x^{2} + \frac{1}{3} x + \frac{1}{36}$

Compare your answer:

If these expressions are written in factored form, each one is the square of a linear expression:

$(x + 3)^{2}$ , $(x - 8)^{2}$ , and $(x + \frac{1}{6})^{2}$ .

Are you ready for more?

Extending Your Thinking

Write each expression in factored form.

$x^{4} - 30 x^{2} + 225$

Compare your answer:

$(x^{2} - 15)^{2}$

$x + 14 \sqrt{x} + 49$

Compare your answer:

$(\sqrt{x} + 7)^{2}$

$5^{2 x} + 6 \cdot 5^{x} + 9$

Compare your answer:

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

Self Check

Which of the following is equivalent to

(x-4)^2

$x^2+8x+16$
$x^2-8x+16$
$x^2+16$
$x^2-16$

Additional Resources

Creating Perfect Square Trinomials

A perfect square trinomial is created by multiplying a binomial by itself.

Example

Write $(x - 2)^{2}$ as an expanded polynomial in standard form:

Step 1 - Multiply the binomial by itself.

$(x - 2) (x - 2)$

Step 2 - Use the distributive property to expand (FOIL).

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

Step 3 - Combine like terms.

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

Try it

Try It: Creating Perfect Square Trinomials

Write $(x + 5)^{2}$ as an expanded polynomial in standard form.

Here is how to expand the polynomial:

Step 1 - Multiply the binomial by itself.
$(x + 5) (x + 5)$

Step 2 - Use the distributive property (FOIL).
$x^{2} + 5 x + 5 x + 25$

Step 3 - Combine like terms.
$x^{2} + 10 x + 25$

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- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
- Publisher/website: OpenStax
- Book title: Algebra 1
- Publication date: Jun 4, 2025
- Location: Houston, Texas
- Book URL: https://openstax.org/books/algebra-1/pages/about-this-course
- Section URL: https://openstax.org/books/algebra-1/pages/9-1-2-recognizing-structure-in-perfect-square-expressions

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