Keep Both Roots of a Shifted Square
A squared expression equal to a positive square can have either positive or negative square-root value.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Solve (x−2)²=9. Candidate answers: (A) x=-1 or x=5 (B) x=-7 or x=11 (C) x=11 (D) x=5 only.
- Independent check inputs: Solve (x−3)²=16. Candidate answers: (A) x=-1 or x=7 (B) x=-13 or x=19 (C) x=19 (D) x=7 only.
Learn the idea
Quadratic equations can be solved by revealing factors, making a square, or undoing a square. Each method exposes structure, and all valid roots must be retained and checked in the original equation. A squared expression equal to a positive square can have either positive or negative square-root value. Undo the shift only after recording both signed possibilities.
Here is the method on the supplied teaching case. Take both square-root possibilities: x−2=±3. Add 2 to each: x=-1 or 5. A square equation can have two real solutions.
The principal square-root symbol itself is nonnegative, but solving z²=b requires z=±√b for positive b. Omitting the negative branch can lose a valid equation solution.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Solve (x−2)²=9. Candidate answers: (A) x=-1 or x=5 (B) x=-7 or x=11 (C) x=11 (D) x=5 only. Solution: Take both square-root possibilities: x−2=±3. Add 2 to each: x=-1 or 5. A square equation can have two real solutions. The final answer is x=-1 or x=5. The principal square-root symbol itself is nonnegative, but solving z²=b requires z=±√b for positive b. Omitting the negative branch can lose a valid equation solution.
Check your reasoning
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Helpful tips
- A squared expression equal to a positive square can have either positive or negative square-root value.
- The principal square-root symbol itself is nonnegative, but solving z²=b requires z=±√b for positive b.
See the answer and explanation
x=-1 or x=7
Take both square-root possibilities: x−3=±4. Add 3 to each: x=-1 or 7. A square equation can have two real solutions.
Take it further
Changed-input task: Solve (x−9)²=100. Candidate answers: (A) x=-1 or x=19 (B) x=-91 or x=109 (C) x=109 (D) x=19 only. Show the mathematical reason, not only the selected answer. Expected result: x=-1 or x=19. Take both square-root possibilities: x−9=±10. Add 9 to each: x=-1 or 19. A square equation can have two real solutions.
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