Checking Solutions Examples in Math

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

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

Concept Recap

Checking solutions means substituting candidate values back into the original condition to verify they satisfy it.

Treat your answer as a hypothesis and test it by substituting back into the original equation to verify.

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: Checking a solution substitutes a candidate value into the original condition and confirms both sides truly match.

Common stuck point: The procedure for checking solutions is the easy part; the trap is checking against a transformed equation instead of the original. Asking "Do I have a candidate value that I plug into the original condition to confirm both sides are equal?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I have a candidate value that I plug into the original condition to confirm both sides are equal?

Worked Examples

Example 1

easy

Check whether

x = 5

is a solution of

2x - 3 = 7

Answer

Yes,

x = 5

is a solution.

First step

Substitute

x = 5

into the left side:

2(5) - 3 = 10 - 3 = 7

Full solution

2
Compare with right side: $7 = 7$ ✓
3
Yes, $x = 5$ is a solution.

Checking a solution means substituting the candidate value and verifying both sides of the equation are equal.

Example 2

medium

Solve

\sqrt{x+3} = x - 3

and check for extraneous solutions.

Example 3

medium

After solving

\sqrt{2x+1}=x-1

, candidates are

x=0

and

x=4

. Check each.

Example 4

medium

Solve

x^2=16

and verify both candidates.

Example 5

hard

Solve

\sqrt{x+5}=x-1

and verify the candidates.

Example 6

hard

For

x^2-7x+10=0

, candidates from factoring are

x=2,5

. Verify both.

Example 7

challenge

Solve

\sqrt{x+4}+\sqrt{x-1}=5

and verify the candidate.

Practice Problems

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

Example 1

easy

x = -2

a solution of

x^2 + x = 2

Example 2

medium

Someone solved

\frac{x}{x-2} = \frac{2}{x-2}

and got

x = 2

. Is this valid?

Example 3

easy

x=3

a solution of

2x+1=7

Example 4

easy

x=5

a solution of

x-2=4

Example 5

easy

Does

x=0

satisfy

3x=0

Example 6

easy

x=2

a solution of

x^2=4

Example 7

easy

x=-2

a solution of

x^2=4

Example 8

easy

x=4

a solution of

\frac{x}{2}=2

Example 9

easy

x=1

a solution of

2x+3=x+5

Example 10

easy

x=9

a solution of

\sqrt{x}=3

Example 11

medium

Both

x=2

and

x=3

are proposed for

x^2-5x+6=0

. Which actually work?

Example 12

medium

After squaring, a student gets

x=4

for

\sqrt{x}=-2

. Is it a valid solution?

Example 13

medium

x=-1

a solution of

\frac{2}{x+1}=3

Example 14

medium

Verify whether

(x,y)=(2,1)

solves the system

x+y=3

and

2x-y=3

Example 15

medium

x=3

a solution of

|x-1|=2

Example 16

medium

A student rounds

\sqrt{2}\approx1.41

and claims

x=1.41

solves

x^2=2

. Does it exactly satisfy it?

Example 17

medium

Does

x=5

satisfy the inequality

2x-3>6

Example 18

medium

For

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

, is

x=1

a valid solution even though the simplified form gives

2

Example 19

medium

(x,y)=(1,2)

a solution of the system

y=2x

and

x+y=3

Example 20

challenge

Find ALL values of

a

for which

x=2

is a solution of

x^2+ax-6=0

, and verify.

Example 21

challenge

Squaring

\sqrt{x+6}=x

gives

x=3

x=-2

. Determine which are valid.

Example 22

challenge

For what value(s) of

x

is the proposed solution

x=4

valid in

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

Example 23

easy

x=4

a solution of

3x-2=10

Example 24

easy

x=-3

a solution of

x^2-9=0

Example 25

easy

x=0

a solution of

5x+2=2

Example 26

easy

x=-5

a solution of

x+5=0

Example 27

medium

x=3

a solution of

\frac{x+1}{x-3}=2

Example 28

medium

Verify: is

(x,y)=(3,4)

a solution of

x^2+y^2=25

Example 29

medium

x=1

a solution of

\log_{10}(x)=0

Example 30

medium

x=-2

a valid solution of

\log(x)=1

Example 31

medium

x=4

a solution of

|x-1|=3

Example 32

medium

After solving, a student claims

x=10

solves

\frac{x}{x-10}=1

. Verify.

Example 33

medium

Verify whether

x=5

satisfies the inequality

3x-4<10

Example 34

hard

For

\frac{1}{x-2}-\frac{1}{x}=\frac{2}{x(x-2)}

, is

x=2

a valid solution?

Example 35

hard

A student squares

\sqrt{x}=x-2

to get

x^2-5x+4=0

, giving

x=1

x=4

. Which check out?

Example 36

hard

(x,y)=(1,1)

a solution of

x^2+y^2=4

Example 37

hard

For

\log_2(x-1)+\log_2(x+1)=3

, a candidate is

x=3

. Verify.

Example 38

challenge

For what value of

b

x=3

a solution of

x^2+bx-15=0

? Verify.

Example 39

challenge

Suppose squaring gives candidates

x=2

and

x=-2

for

\sqrt{4-x^2}=x

. Which are valid?

← Back to Checking Solutions Practice Mode

Related Concepts

Evaluation Solution Concept Equivalence

Background Knowledge

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

evaluationsolution conceptequivalence