Evaluation Examples in Math

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

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

Concept Recap

The process of calculating the numerical value of a mathematical expression by substituting specific numbers for each variable and then performing the indicated operations following the order of operations (PEMDAS/BODMAS).

Plug in the number and compute: if $x = 3$ , then $2x + 1 = 2(3) + 1 = 7$ .

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: Evaluation substitutes given numbers for the variables and works out the single resulting value.

Common stuck point: The procedure for evaluation is the easy part; the trap is ignoring order of operations after substituting. Asking "Are the variable's values given so I just substitute and compute one number?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the variable's values given so I just substitute and compute one number?

Worked Examples

Example 1

easy

Evaluate

5x - 3

when

x = 4

Answer

17

First step

Substitute

x = 4

5(4) - 3

Full solution

2
Multiply: $20 - 3$ .
3
Subtract: $17$ .

Evaluation means replacing the variable with a given value and computing the result using the order of operations.

Example 2

medium

Evaluate

\frac{a^2 - b^2}{a + b}

when

a = 5

and

b = 3

Example 3

medium

Evaluate

h(t) = -16t^2 + 64t + 5

t = 2

Example 4

medium

Evaluate

\frac{x^2 + 2x}{x - 1}

x = 3

Example 5

medium

Evaluate

f(g(x))

x = 2

when

f(x) = x + 3

and

g(x) = x^2

Example 6

challenge

Given

f(x) = x^2 + bx + c

with

f(0) = 4

and

f(1) = 9

, evaluate

f(-2)

Practice Problems

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

Example 1

easy

Evaluate

-2x + 10

when

x = -3

Example 2

hard

f(x) = x^3 - 2x

2x^2-3x+1

x=-2

Example 13

medium

Evaluate

\sqrt{x^2+9}

x=4

Example 14

medium

C=\frac{5}{9}(F-32)

, evaluate

C

F=212

Example 15

medium

Evaluate

3(x-2)^2+4

x=5

Example 16

medium

Evaluate

\frac{ab}{a+b}

a=6,\ b=3

Example 17

challenge

For

f(x)=x^2-4x

, find the value of

x

where evaluating gives the same result as at

x=1

, other than

x=1

Example 18

challenge

Evaluate

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

x=3

, then explain what value it approaches as

x\to 3

Example 19

challenge

A formula gives

P=2(\ell+w)

. If evaluating at

\ell=w=s

yields

P=20

Evaluate

5x - 2y + 1

x = 3,\ y = -2

Example 28

medium

g(x) = \frac{x^2 - 1}{x + 1}

, evaluate

g(4)

Example 29

medium

Evaluate

2x^3 - x + 4

x = -1

Example 30

medium

f(x) = \sqrt{2x + 3}

, find

f(11)

Example 31

medium

A = \frac{1}{2}bh

, find

A

when

b = 7

and

h = 12

Example 32

medium

Evaluate

|2x - 9|

x = 2

Example 33

medium

Evaluate

3xy - x^2

x = 2,\ y = 5

Example 34

medium

f(x) = 2^x

, find

f(5)

Example 35

hard

Evaluate

\frac{x^3 + 8}{x + 2}

x = -2

, then state the limit as

x \to -2

Example 36

hard

f(x) = x^2 - 6x + 11

, find the value of

x

that makes

f(x) = 2

Example 37

hard

Evaluate

f(x+h) - f(x)

for

f(x) = x^2

, then divide by

h

Example 38

hard

f(x) = \frac{2x - 1}{x + 3}

, find

f(-1) + f(1)

Example 39

challenge

f(x) = ax + b

and

f(1) = 5

f(3) = 11

, find

f(10)

Example 40

challenge

Evaluate the sum

\sum_{k=1}^{4} (k^2 - k)

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Related Concepts

Expressions Order of Operations Checking Solutions

Background Knowledge

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

expressionsorder of operations