Choose a representation
Before squaring a square-root equation, require its other side to be nonnegative. Use two signed branches for a positive absolute-value distance. Use quadratic structure when a sum or height is requested.
Worked example
What is the real solution of √(x+12)=x? The right side requires x≥0 before squaring. Squaring gives (x−4)(x+3)=0; reject the negative candidate in the original equation. Check x=4: √(4+12)=4; candidate -3 fails because a square root cannot equal that negative value. The requested value is 4.
Contrast a wrong operation
In that example, -3 follows this error: Keep the extraneous negative candidate introduced by squaring. The correct requested value is 4. Explain which quantity the mistaken method is actually calculating.
Connect a second form
For 3 x²−12 x+6=0, let r and s be its roots, counted with multiplicity. What is r+s? Use root sum equal to negative linear coefficient divided by leading coefficient. The nonmonic leading coefficient must remain in the denominator. Substitute the given numbers: 12 ÷ 3 = 4. The requested value is 4.
Transfer and check
What is the real solution of √(x+20)=x? Pause before checking. Answer: 5. The right side requires x≥0 before squaring. Squaring gives (x−5)(x+4)=0; reject the negative candidate in the original equation. Check x=5: √(5+20)=5; candidate -4 fails because a square root cannot equal that negative value.
Boundaries of the method
Squaring can introduce an extraneous root. The root sum of a nonmonic quadratic divides by its leading coefficient. A maximum height is an output, not the time when it occurs. This lesson samples a PSAT/NMSQT objective; its examples are original and do not claim official question status or full coverage.
What is the real solution of √(x+12)=x?
- The right side requires x≥0 before squaring.
- Squaring gives (x−4)(x+3)=0; reject the negative candidate in the original equation.
- Check x=4: √(4+12)=4; candidate -3 fails because a square root cannot equal that negative value.
- The requested value is 4.
Compare the other answer choices
Report the constant under the radical as the solution.
Report the square of the valid input rather than the input.
Keep the extraneous negative candidate introduced by squaring.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications