Reject a Candidate with the Wrong Original Sign

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Solve √(x+2)=−3 over the reals. Candidate answers: (A) No real solution (B) x=5 (C) x=7 (D) x=−11.
• Independent check inputs: Solve √(x+3)=−4 over the reals. Candidate answers: (A) No real solution (B) x=13 (C) x=7 (D) x=−19.

Learn the idea

Root expressions can be simplified through perfect-power factors and rational exponents. Squaring equations can introduce extraneous candidates, while complex arithmetic extends the number system using an imaginary unit with square negative one. A principal square root cannot equal a negative real number. Squaring both sides may erase that sign contradiction and produce an algebraic candidate that fails in the original equation.

Here is the method on the supplied teaching case. The principal square root is nonnegative, whereas −3<0. Squaring produces x=7, but it does not satisfy the original sign condition. Checking rejects this extraneous candidate.

Squaring can enlarge a solution set, so every candidate needs substitution back into the original. Checking only the squared equation would miss the original nonnegative-root requirement and retain an extraneous answer.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Solve √(x+9)=−10 over the reals. Candidate answers: (A) No real solution (B) x=19 (C) x=91 (D) x=−109. Show the mathematical reason, not only the selected answer. Expected result: No real solution. The principal square root is nonnegative, whereas −10<0. Squaring produces x=91, but it does not satisfy the original sign condition. Checking rejects this extraneous candidate.

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