A.9(F)·Algebra II·Square Root FunctionsReadiness

Solving Square Root Equations

Solve square-root equations and check proposed solutions in the original equation.

Students solve square-root equations by isolating the radical, squaring both sides, and checking every result in the original equation to reject extraneous solutions.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice45 minutes

Learn the concept

Isolate, then square

Get the square-root expression alone on one side first. Then square both sides to remove it: $\sqrt{2x-1}=5$ becomes $2x-1=25$ . Squaring before isolating creates harder expressions and more errors.

Square the whole side

When the other side is a binomial, square the entire binomial: $(x-1)^2=x^2-2x+1$ . Rearrange the result to equal zero and solve, often by factoring.

Check for extraneous solutions

The symbol $\sqrt{\phantom{x}}$ means the nonnegative root. A value may solve the squared equation but make the original equation say a positive number equals a negative one. Reject such extraneous values.

Worked example

Solve $\sqrt{x+5}=x-1$ .

Square both sides: $x+5=x^2-2x+1$ .
Rearrange: $x^2-3x-4=0$ , so $(x-4)(x+1)=0$ .
Check $x=4$ : $\sqrt{9}=3$ and $4-1=3$ . It works.
Check $x=-1$ : $\sqrt{4}=2$ , but $-1-1=-2$ . It is extraneous.

$x=4$

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Solving Contextual Square Root Equations

UT IPSI · Kid2Kid / Texas Education Agency

Solve square-root equations that model real-world situations and interpret the solutions.

Watch on YouTube

Texas public school video

Algebra 2 Formulating Square Root Functions

UT IPSI · Kid2Kid / Texas Education Agency

Write square-root functions from contexts before solving the related equations.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. Solve $\sqrt{2x-1}=5$ .
Show answer and reasoning
$x=13$
Squaring gives $2x-1=25$ , so $x=13$ . Check: $\sqrt{25}=5$ .
2. Solve $x+\sqrt{x}=12$ by testing perfect squares.
Show answer and reasoning
$x=9$
$9+\sqrt{9}=9+3=12$ . The left side increases as x increases, so no other value works.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 9 · 1 point

The sum of a number, $n$ , and its square root can be represented by $y = n + \sqrt{n}$ . If $y = 20$ , which of the following is true?

$n = 16$
$n = 4$
$n = 16$ and $n = 25$
$n = 4$ and $n = 5$

Show STAAR answer and explanation

$n = 16$

Let $u=\sqrt{n}$ , so $u^2+u=20$ and $(u+5)(u-4)=0$ . Since $u=\sqrt{n}\ge0$ , only $u=4$ is valid, giving $n=16$ . Check: $16+4=20$ .

$n = 4$ : Then $n+\sqrt{n}=4+2=6$ , not 20; 4 is the square root of the answer, not n itself.

$n = 16$ and $n = 25$ : Squaring introduces the extraneous value 25: $25+\sqrt{25}=30$ , not 20.

$n = 4$ and $n = 5$ : These are possible values of the auxiliary variable or its rejected negative root, not valid values of n in the original equation.

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Archived released STAAR Algebra II EOC · 2013 · Item 49 · 1 point

The average speed of sound in air at $t$ degrees Celsius can be found using the function $s = 20.05\sqrt{t + 273.15}$ . The speed, $s$ , is measured in meters per second. If sound is traveling in air at an average speed of 356.5 meters per second, which temperature is closest to the value of $t$ ?

76°C
503°C
43°C
309°C

Show STAAR answer and explanation

43°C

Divide: $\sqrt{t+273.15}=\frac{356.5}{20.05}\approx17.78$ . Square: $t+273.15\approx316.2$ , so $t\approx43$ °C.

76°C: $20.05\sqrt{76+273.15}\approx374.6$ meters per second.

503°C: $20.05\sqrt{503+273.15}\approx558.6$ meters per second.

309°C: $20.05\sqrt{309+273.15}\approx483.8$ meters per second.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Isolate a square-root expression before squaring.
Square binomial expressions correctly when solving.
Check proposed solutions and identify extraneous solutions.

Instructional sequence

1
Engage
5 min
Ask students to find a number that, added to its square root, equals 12. Let them reason with perfect squares.
2
Direct instruction
10 min
Model the isolate–square–solve–check routine and explain why squaring can create extra answers.
3
Guided practice
12 min
Solve equations with the radical already isolated, then equations that require isolating it first.
4
Error analysis
12 min
Analyze student work that keeps an extraneous solution or squares terms separately. Write the correction.
5
Closure
6 min
Complete the exit ticket, including the check.

Misconceptions to address

““

(x-1)^2=x^2+1

.””

Teacher move: Square the whole binomial: $(x-1)^2=x^2-2x+1$ . Use an area model if needed.

““Every solution to the squared equation solves the original.””

Teacher move: Squaring can introduce values that make a principal square root equal a negative number. Substitute every value into the original equation.

Materials

•Equation-solving organizer with a check column
•Perfect-square reference list
•Calculator

Teaching tips

Make the check column mandatory on every problem, not optional.
Remind students that the principal square root is never negative.
Use perfect-square reasoning to estimate before solving algebraically.

Exit ticket · 3–5 min

Solve $\sqrt{x+7}=x-5$ and check your answers.

Show expected answer

Squaring gives $x^2-11x+18=0$ , so $x=9$ or $x=2$ . Check: $\sqrt{16}=4=9-5$ works; $\sqrt{9}=3\ne -3$ , so 2 is extraneous. The solution is $x=9$ .

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Isolate, then square

Square the whole side

Check for extraneous solutions

Worked example

Watch the lesson

Algebra 2 Solving Contextual Square Root Equations

Algebra 2 Formulating Square Root Functions

Try it yourself

1. Solve 2x−1=5\sqrt{2x-1}=52x−1​=5.

2. Solve x+x=12x+\sqrt{x}=12x+x​=12 by testing perfect squares.

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 9 · 1 point

Archived released STAAR Algebra II EOC · 2013 · Item 49 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Error analysis

Closure

Misconceptions to address

1. Solve $\sqrt{2x-1}=5$ .

2. Solve $x+\sqrt{x}=12$ by testing perfect squares.