Rewriting Expressions Examples: 43 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rewriting Expressions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Transforming an algebraic expression into a different but mathematically equivalent form to reveal new information.

$2(x + 3)$ and $2x + 6$ look different but are the same—rewriting shows this.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Rewriting expressions swaps one form for an equal form to expose information the old form hid.

Common stuck point: The procedure for rewriting expressions is the easy part; the trap is distributing to only the first term. Asking "Are these two expressions equal at every value of the variable, just written differently?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are these two expressions equal at every value of the variable, just written differently?

Worked Examples

Example 1

easy

Rewrite

x^2 - 25

in factored form.

Answer

(x + 5) (x - 5)

First step

Recognize as a difference of squares:

x^2 - 5^2

Full solution

2
Apply the pattern: $a^2 - b^2 = (a+b)(a-b)$ .
3
Result: $(x+5)(x-5)$ .

Rewriting an expression means transforming it into an equivalent form. The difference of squares pattern is one of the most useful rewriting tools in algebra.

Example 2

medium

Rewrite

\frac{x^2 - 4}{x - 2}

in simplified form.

Example 3

medium

Rewrite

(x+3)^2

in expanded form.

Example 4

medium

Expand

(x-2)(x+5)

Example 5

medium

Rewrite

x^2+6x

by completing the square.

Example 6

hard

Factor

x^2-5x-14

Example 7

hard

Expand and simplify

(x+1)(x-1)(x+2)

Example 8

hard

Rewrite

x^2-10x+7

by completing the square.

Example 9

challenge

Factor

x^4-5x^2+4

completely.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Rewrite

3(x + 4)

in expanded form.

Example 2

medium

Rewrite

4x + 12

in factored form.

Example 3

easy

Rewrite

3(x+4)

without parentheses.

Example 4

easy

Are

2x+6

and

2(x+3)

equivalent?

Example 5

easy

Rewrite

x+x+x

as a single term.

Example 6

easy

Rewrite

\frac{6x}{3}

in simplest form.

Example 7

easy

Write

5x+2x

as one term.

Example 8

easy

Rewrite

x^2\cdot x^3

using one exponent.

Example 9

easy

Rewrite

2(a+b)+3(a+b)

as a single product.

Example 10

easy

Rewrite

-(x-4)

without parentheses.

Example 11

medium

Rewrite

\frac{x^2-9}{x-3}

in simpler form (assume

x\ne 3

Example 12

medium

Rewrite

x^2+6x+9

as a squared binomial.

Example 13

medium

Rewrite

\frac{1}{2}x+\frac{1}{4}x

as a single term.

Example 14

medium

Rewrite

2x+10

as a product of a constant and a binomial.

Example 15

medium

Rewrite

(x+1)^2

without parentheses.

Example 16

medium

A rectangle has area

x^2+5x

. Rewrite it as length times width.

Example 17

medium

Rewrite

\frac{3}{x}+\frac{2}{x}

as a single fraction.

Example 18

medium

Rewrite

50\cdot 99

using

99=100-1

to compute mentally.

Example 19

medium

Rewrite

\sqrt{50}

in simplest radical form.

Example 20

challenge

Rewrite

x^2+8x

to reveal its minimum value form (complete the square).

Example 21

challenge

Show

\frac{1}{x}-\frac{1}{x+1}

rewrites to a single fraction, then evaluate at

x=4

Example 22

challenge

Rewrite

a^2-b^2

to explain why

1000^2-999^2=1999

Example 23

easy

Expand

5(2x-3)

Example 24

easy

Combine like terms:

4x+3-x+8

Example 25

easy

Simplify

\tfrac{15x^2}{5x}

for

x\ne 0

Example 26

easy

Expand

-2(3x+4)

Example 27

medium

Factor

x^2+7x+12

Example 28

medium

Simplify

\tfrac{x^2-9}{x+3}

for

x\ne -3

Example 29

medium

Factor

2x^2-8

Example 30

medium

Combine:

\tfrac{1}{x}+\tfrac{2}{x}

Example 31

hard

Simplify

\tfrac{x^2+5x+6}{x+2}

for

x\ne -2

Example 32

hard

Factor

3x^2+10x+8

Example 33

hard

Simplify

\tfrac{x^2-4}{x^2-x-6}

and state restrictions.

Example 34

hard

Expand

(a+b+c)^2

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Related Concepts

Expressions Distributive Property Factoring

Background Knowledge

These ideas may be useful before you work through the harder examples.

expressionsdistributive property