Factor a difference of squares

Ontario Grade 10 Mathematics

Quadratic Relations

Recognize two square terms being subtracted and rewrite them as a pair of factors

Factoring means rewriting an expression as a product of simpler expressions. For example, multiplication can turn two factors into a polynomial, and factoring reverses that process. This lesson focuses on one pattern: a difference of squares. The word difference means subtraction. A square is the result of multiplying a number or term by itself. Once you recognize two squared terms with a subtraction sign between them, you can use a reliable pair of factors.

What you will learn

Recognize when an expression is a difference of squares.
Explain why the difference-of-squares pattern works.
Factor a difference of squares using the correct signs.
Check a factorization by expanding the factors.

1. Build from familiar ideas

Before factoring, recall how brackets are multiplied. The distributive property says that a factor outside brackets multiplies each term inside. For example, expanding

(x+3)(x-3)

gives

x^2-3x+3x-9

. The middle terms cancel, leaving

x^2-9

.

That cancellation is the key. The two factors have the same terms, but one uses addition and the other subtraction. When multiplied, the matching middle terms cancel. The result is a subtraction of two squares.

To factor, work backward from that result. Look for two terms that are being subtracted, and ask whether each term is a perfect square. A perfect square can be written as one expression multiplied by itself. For instance,

x^2

is the square of

x

, and

25

is the square of

5

.

Factoring reverses expansion.
A perfect square is an expression multiplied by itself.
The subtraction sign between the terms matters.

2. The difference-of-squares pattern

Suppose an expression has the form

A^2-B^2

. Here,

A

and

B

stand for expressions whose squares appear in the original expression. The expression factors into one sum and one difference:

(A+B)(A-B)

.

This pattern works because expanding those factors gives

A^2-AB+AB-B^2

. The middle terms,

-AB

and

+AB

, add to zero. What remains is

A^2-B^2

.

Use this pattern only when there are two square terms and they are separated by subtraction. A sum such as

x^2+16

is not a difference of squares, even though both terms are squares. The pattern also does not apply directly if one of the terms is not a perfect square.

A helpful recognition routine is to identify the first square, identify the second square, and then put their square roots into a pair of brackets. Use a plus sign in one bracket and a minus sign in the other. The order of the two brackets can be reversed without changing the product.

(A^2-B^2)=(A+B)(A-B)

The pattern uses subtraction between two perfect squares.
The factors contain the same two terms, once with addition and once with subtraction.
Check that the middle terms cancel when you expand.

3. A reliable factoring routine

Start by checking the sign between the terms. If it is subtraction, inspect each term to see whether it is a square. Then write each term as a square so the roots are visible. Apply the pattern, and finally expand the factors to check your work.

The roots must include any number or variable factor. For example, the square root of

9x^2

is

3x

, because

(3x)(3x)=9x^2

. The square root of

25y^2

is

5y

. Leaving out the numerical factor would make the proposed factors expand to the wrong expression.

This routine is about recognizing and using one pattern. It does not require solving an equation. If the task is to factor an expression, the goal is to show an equivalent product, not to find values of a variable.

Check subtraction first, then identify both square roots.
Include numerical and variable factors in each square root.
Expand the final product to verify it matches the original expression.

4. Check by expanding

Expansion is a useful check because it reverses the factoring step. Multiply each pair of terms. The two middle products should be equal in size and opposite in sign, so they cancel. The remaining squared terms should match the original expression.

If the middle terms do not cancel, or if a coefficient is incorrect, revisit the square roots. Also check that the original expression had subtraction rather than addition. A correct factorization must reproduce every term and sign when expanded.

The product of the proposed factors must equal the original expression.
Opposite middle terms cancel.
A quick expansion can reveal a missing coefficient or incorrect sign.

Worked example

Factor a numerical square and a variable square

Factor

x^2-64

.

Recognize the two squares
The expression has subtraction. The first term is the square of $x$ , and $64$ is the square of $8$ .
$x^2-64=x^2-8^2$
Use the pattern
Use the two square roots, $x$ and $8$ , in a sum and a difference. This gives factors whose middle products cancel when expanded.
$(x+8)(x-8)$
Verify by expanding
The cross-products are $-8x$ and $+8x$ , so they cancel. The remaining terms match the original expression.
$(x+8)(x-8)=x^2-64$

Answer:

(x+8)(x-8)

Check: Expanding gives

x^2-8x+8x-64=x^2-64

.

Worked example

Factor expressions with coefficients and two variables

Factor

16m^2-49n^2

.

Rewrite each term as a square
There is subtraction. Since $(4m)^2=16m^2$ and $(7n)^2=49n^2$ , the square roots are $4m$ and $7n$ .
$16m^2-49n^2=(4m)^2-(7n)^2$
Apply the pattern
Place the roots in two brackets, using addition in one and subtraction in the other. Keeping both numerical factors is important for the product to be correct.
$(4m+7n)(4m-7n)$
Verify by expanding
The middle products are $-28mn$ and $+28mn$ , which cancel. The squared terms left over are the two terms from the original expression.
$(4m+7n)(4m-7n)=16m^2-49n^2$

Answer:

(4m+7n)(4m-7n)

Check: Expansion gives

16m^2-28mn+28mn-49n^2=16m^2-49n^2

.

Common mistakes and how to avoid them

Using the pattern on a sum of squares, such as

x^2+25

.

Correction: The pattern in this lesson requires subtraction between the two squares.

Writing two factors with the same sign, such as

(x+8)(x+8)

for

x^2-64

.

Correction: Use one sum and one difference so that the middle products cancel.

Forgetting the numerical part of a square root, such as treating the square root of

16m^2

as

m

.

Correction: Check by squaring the proposed root. Since

(4m)^2=16m^2

, use

4m

.

Assuming the factors are correct without checking.

Correction: Expand the brackets. The result must match the original expression term by term.

Lesson summary

A difference of squares has two perfect-square terms separated by subtraction.
Write the square roots in two factors, with one sum and one difference.
Expand the factors to confirm that the middle terms cancel and the original expression returns.

Check your understanding

Question 1

Which expression is a difference of squares?

$p^2+36$
$p^2-36$
$p^2-6$
$p^2+6$

Show answer and explanation

p^2-36

p^2

and

36=6^2

are perfect squares, and they are separated by subtraction.

Question 2

What is the factorization of

y^2-100

?

$(y+10)(y-10)$
$(y-10)(y-10)$
$(y+100)(y-1)$
$(y+10)(y+10)$

Show answer and explanation

(y+10)(y-10)

The square roots are

y

and

10

, so use one sum and one difference. Expanding the factors gives

y^2-100

.

Question 3

What is the square root of

25r^2

?

$5r$
$25r$
$5r^2$
$r$

Show answer and explanation

5r

(5r)^2=25r^2

, so the square root needed for the pattern is

5r

.

Key terms

Factor: One of the expressions multiplied together to make a product.
Factoring: Rewriting an expression as a product of factors.
Perfect square: A number or expression made by multiplying an expression by itself.
Difference: The result of subtracting one quantity from another.
Expand: Multiply factors and use the distributive property to write the product as a sum or difference of terms.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic Q5. It is a study resource, not an official curriculum publication.

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