Principal Square Roots and Equation Solutions

SAT Glossary

The rule separating x^2=49 (two solutions, ±7) from √(49) (one number, 7): the ± is supplied by you when you apply a square root to solve an equation, never by the symbol, which has already committed to the positive branch. Items offering '±7' for √(49) and '7' for x^2=49 test the same distinction from opposite sides, and both appear. Desmos will mislead you here — it returns the principal root and will not remind you of the second solution.

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for $\\sqrt{49}$ and '7' for $x^2=49$ test the same distinction from opposite sides, and both appear. Desmos will mislead you here — it returns the principal root and will not remind you of the second solution.","section":"Mathematics","source":"KomFi SAT vF - Chapter 38: Radicals and Rational Exponents","necessity":4,"learnMore":"This distinguishes two situations that look similar but behave differently: solving the equation $x^2=49$ by taking a square root gives you TWO solutions, $\\pm7$ (both $7^2=49$ and $(-7)^2=49$ are true) — and that plus-or-minus symbol is something YOU have to add yourself when you take a square root to SOLVE an equation. But the expression $\\sqrt{49}$ all by itself, with no equation to solve, is defined to mean just ONE single number: positive 7 — the radical symbol has already committed to only the positive branch, no exceptions. The test likes to offer '$\\pm7

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as a wrong answer for plain $\\sqrt{49}$, and offer just '7' as a wrong (incomplete) answer for the equation $x^2=49$ — testing the exact same distinction from both directions at once. Watch out for graphing calculators like Desmos here too: they'll only ever show you the principal (positive) root and won't remind you that a second solution exists.","aliases":[],"topic":"Course Foundations","priority":"specialized","learningStage":"intermediate","prerequisiteIds":[],"questionCount":0,"curriculumBasis":{"revision":"current-sat-2026","objective":"Course Foundations: You supply the $\\pm$","source":"https://satsuite.collegeboard.org/practice/content-domains"},"id":"Principal Square Roots and Equation Solutions","courseId":"sat","publicSeo":true,"seoSlug":"principal-square-roots-and-equation-solutions-4w6n7i"}

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KomFi: Test Prep Made Easy — free adaptive practice for GMAT, GRE, SAT, ACT, National Real Estate Exam, Investment Banking, and finance with full explanations.

KomFi Academy is free GMAT prep and personalized GMAT help built as a training platform: 75,000+ practice questions, 26,500+ flashcards, on-demand video lectures, podcasts, and 4K slide decks. Flagship tracks: Free GMAT Prep, Free GMAT Resources, National Real Estate Exam Prep, Investment Banking Prep, Finance Prep, GRE, SAT, ACT, LSAT, MCAT, Financial Accounting, Private Equity, Private Credit, and Quantitative Finance.

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