Radical Equations

Explore isolate before squaring, extraneous solutions, real-domain checks.

Lesson 5 of 53 parts Practice Flashcards

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Isolate Before Squaring

Objective: Explain isolate before squaring and use it to solve or interpret a problem.

Move other terms before raising both sides to a power; this makes the radical alone and helps avoid algebraic errors. Squaring a sum creates cross terms.

Worked examples

Try itRadical Equations — practice 1.1: Solve √(x+4)=5 for x.Show answer

Step 1: Square both sides: x+4=25. Step 2: Subtract 4: x=21. Step 3: Check the original principal root is 5.

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Isolate Before Squaring

Objective: Explain isolate before squaring and use it to solve or interpret a problem.

Move other terms before raising both sides to a power; this makes the radical alone and helps avoid algebraic errors. Squaring a sum creates cross terms.

Worked examples

Try itRadical Equations — practice 1.1: Solve √(x+4)=5 for x.Show answer

Step 1: Square both sides: x+4=25. Step 2: Subtract 4: x=21. Step 3: Check the original principal root is 5.

Extraneous Solutions

Objective: Explain extraneous solutions and use it to solve or interpret a problem.

Squaring can turn a false negative-root equality into a true polynomial equality. Every candidate must satisfy the original radical equation and real domain. In the unsquared original equation.

Worked examples

Try itRadical Equations — practice 2.1: How many real solutions does √x=−3 have?Show answer

Step 1: The principal square root is nonnegative. Step 2: The right side is negative. Step 3: No real x satisfies the original equation.

Real-Domain Checks

Objective: Explain real-domain checks and use it to solve or interpret a problem.

Even-root radicands must be nonnegative, and isolated principal roots cannot equal a negative number. State these conditions before solving. No, its principal value is nonnegative.

Worked examples

Try itRadical Equations — practice 3.1: How many real solutions does √x=−4 have?Show answer

Step 1: The principal square root is nonnegative. Step 2: The right side is negative. Step 3: No real x satisfies the original equation.

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