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A.9(D)·Algebra II·Square Root FunctionsSupporting

Square Root Equations

Solve square root equations algebraically and check for extraneous solutions.

Students solve square-root equations by isolating the radical, squaring both sides, and checking proposed solutions in the original equation.

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 tutorAll Algebra II released practice45 minutes

Learn the concept

Respect the principal root

A real square root requires a nonnegative radicand and has a nonnegative value. An equation such as x+2=−3\sqrt{x+2}=-3 has no real solution.

Isolate before squaring

For 2+a−bx=c2+\sqrt{a-bx}=c, subtract 2 first. Then square both entire sides and solve the resulting equation.

Check for extra answers

A candidate can satisfy the squared equation without satisfying the original. Substitute into the original radical and compare both sides, including their signs.

Worked example

Solve 3+14−4n=93+\sqrt{14-4n}=9.

  1. Subtract 3: 14−4n=6\sqrt{14-4n}=6.
  2. Square both sides: 14−4n=3614-4n=36.
  3. Solve: −4n=22-4n=22, so n=−112=−5.5n=-\frac{11}{2}=-5.5.
  4. Check: 3+14−4(−5.5)=3+36=93+\sqrt{14-4(-5.5)}=3+\sqrt{36}=9.

n=−5.5n=-5.5

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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 and check whether proposed solutions are valid.

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TEKSForge practice · unscored

  1. 1. Solve 7−3x=4\sqrt{7-3x}=4.

    Show answer and reasoning

    x=−3x=-3

    7−3x=167-3x=16 gives x=−3x=-3. Substitution gives 16=4\sqrt{16}=4.

  2. 2. Solve x+6=−2\sqrt{x+6}=-2.

    Show answer and reasoning

    No real solution.

    A principal square root cannot be negative. Squaring would give x = −2, but the original left side would be 2, not −2.

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 31 · 1 point

What value of nn makes 10−3n=7\sqrt{10 - 3n} = 7 true? Record your answer.

Show STAAR answer and explanation

-13

Square both sides: 10−3n=4910-3n=49. Then −3n=39-3n=39, so n=−13n=-13. Check: 10−3(−13)=49=7\sqrt{10-3(-13)}=\sqrt{49}=7.

13: With n=13n=13, 10−39=−2910-39=-29, which has no real square root.

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Learning objectives

By the end of the lesson, students can:

  • Determine restrictions imposed by a real square root.
  • Isolate and square a radical expression correctly.
  • Reject extraneous solutions through substitution.

Instructional sequence

  1. 1

    Engage

    5 min

    Compare squaring a positive number and its negative to explain lost sign information.

  2. 2

    Direct instruction

    10 min

    Model isolating the radical and solving a linear equation after squaring.

  3. 3

    Guided practice

    12 min

    Solve equations with negative variable coefficients inside radicals.

  4. 4

    Error analysis

    12 min

    Check a proposed solution that makes the isolated radical equal a negative number.

  5. 5

    Closure

    6 min

    Students solve and check the exit ticket.

Misconceptions to address

““Squaring means multiplying both sides by 2.””

Teacher move: Squaring raises each entire side to the second power.

““Every solution after squaring is valid.””

Teacher move: Squaring loses sign information. Check every candidate in the original equation.