Solve: √(x + 3) = 5

The solution x = 22 emerges when both sides of the equation √(x + 3) = 5 are squared to eliminate the radical, yielding x + 3 = 25, then solving for x by subtracting 3 to obtain x = 22, with verification confirming √(22 + 3) = √25 = 5 satisfies the original equation without extraneous solutions introduced by squaring.

A. 2

Substituting x = 2 yields √(2 + 3) = √5 ≈ 2.236 ≠ 5, failing verification dramatically. The error likely originates from solving x + 3 = 5 (omitting the squaring step) to get x = 2 revealing fundamental misunderstanding that radical equations require exponentiation to eliminate roots rather than direct operand isolation. Students might recognize 5 appears in the equation and execute x + 3 = 5 through visual pattern matching without recognizing the square root operation requires squaring both sides for elimination. Another plausible pathway involves solving √x + 3 = 5 (misplaced radical) to get √x = 2 → x = 4 then adjusting to 2 through unstructured correction. The magnitude disparity (√5 ≈ 2.236 versus 5) reveals complete procedural deficiency in radical equation solving students lack awareness that inverse operations must counter the outermost function (squaring counters square root) before addressing inner operations, a foundational gap requiring explicit function composition analysis with visual scaffolding showing equation structure layers to build systematic solution sequencing preventing operation omission errors that produce mathematically invalid solutions with no verification pathway to recovery.

B. 22

This value correctly satisfies the equation through systematic radical elimination with comprehensive verification. Squaring step: [√(x + 3)]² = 5² → x + 3 = 25. Isolation step: x = 25 - 3 = 22. Verification substitution: √(22 + 3) = √25 = 5 ✓. Domain verification: x + 3 = 25 ≥ 0, satisfying square root domain requirement. Extraneous solution check: squaring can introduce extraneous solutions, but verification confirms validity. Alternative pathway: recognizing 5² = 25 implies radicand must equal 25, so x + 3 = 25 → x = 22. This solution demonstrates comprehensive mastery of radical equation solving including inverse operation application (squaring to eliminate square root), algebraic isolation maintaining equality balance, domain awareness for radical expressions, and mandatory verification to detect extraneous solutions introduced by non-bijective operations integrated competencies essential for calculus (integration requiring radical manipulation), physics (kinematic equations with square roots), engineering (structural load formulas), and computer science (algorithmic constraint solving) where accurate radical equation solution determines valid parameter ranges and physical feasibility across contexts requiring precise root elimination with solution validation.

C. 25

Substituting x = 25 yields √(25 + 3) = √28 ≈ 5.291 ≠ 5, exceeding the target by approximately 0.291. The error likely originates from solving x + 3 = 25 correctly to get x = 22 but then mis-transcribing as 25 through digit substitution influenced by the 25 appearing in the intermediate step. Students might execute squaring correctly to obtain x + 3 = 25 but then incorrectly conclude x = 25 by omitting the subtraction step treating the equation as already solved for x rather than requiring isolation. Another plausible pathway involves solving √x + 3 = 5 → √x = 2 → x = 4 then scaling upward to 25 through unstructured adjustment. The value 25 also equals 5² suggesting students recognized squaring was required but applied it to the solution rather than the equation, executing x = 5² = 25 without incorporating the +3 term. This distractor effectively identifies learners with incomplete solution sequencing who execute partial steps correctly but omit critical isolation operations or confuse intermediate values with final solutions a transitional deficiency requiring explicit step labeling protocols ("Step 1: Square both sides", "Step 2: Isolate variable") with intermediate result recording to prevent conflation between equation transformations and solution values that compromises solution accuracy in increasingly complex radical equation contexts.

D. 28

This value yields √(28 + 3) = √31 ≈ 5.568 ≠ 5, exceeding the target by approximately 0.568. The error likely originates from solving x + 3 = 25 correctly but then adding 3 instead of subtracting (x = 25 + 3 = 28) revealing inverse operation confusion during isolation where students apply the same operation (+3) rather than its inverse (-3) to eliminate the constant term. Students might execute x + 3 = 25 and recognize 3 must be "removed" but incorrectly add 3 to both sides (x + 6 = 28) then misread x = 28. Another plausible pathway involves calculating 5² = 25 then adding the radicand constant 3 to get 28 without equation context. The value 28 also equals 25 + 3 suggesting direct arithmetic combination of intermediate values without algebraic reasoning. This distractor proves particularly instructive for diagnosing inverse operation deficiencies during equation solving where learners fail to recognize that eliminating a term requires applying its inverse operation to both sides a fundamental algebraic principle extending beyond radical equations to all equation types. This deficiency requires explicit emphasis on operation-inverse pairing (addition ↔ subtraction, multiplication ↔ division, exponentiation ↔ roots) with visual balance scale models demonstrating how inverse operations restore equilibrium to isolate variables, building robust solution sequencing habits preventing operation direction errors that produce systematically biased solutions with verification failure revealing the procedural flaw.

Conclusion

The solution x = 22 emerges through rigorous application of inverse operations to eliminate the square root (squaring both sides), followed by algebraic isolation of the variable and mandatory verification to ensure the solution satisfies the original equation without extraneous values introduced by the squaring operation. This problem reinforces critical equation-solving competencies essential across mathematical domains: recognizing the outermost function requiring elimination through inverse operations, executing squaring to remove square roots while acknowledging potential extraneous solution introduction, applying inverse operations correctly during isolation (subtraction to eliminate addition), maintaining domain awareness for radical expressions (radicand ≥ 0), and verifying all solutions in the original equation to detect extraneous values. Mastery of these integrated skills proves indispensable for calculus (solving equations arising from derivatives/integrals), physics (manipulating formulas with radicals like kinetic energy), engineering (solving design constraint equations), and computer algebra systems where accurate radical equation solution determines valid parameter ranges and physical feasibility. The distractors strategically target pervasive error patterns including operation omission (failing to square), solution sequencing errors (omitting isolation step), inverse operation confusion (adding instead of subtracting), and intermediate value conflation highlighting the necessity of explicit stepwise protocols with operation labeling and mandatory verification before final answer acceptance to prevent undetected errors that compromise solution validity in increasingly complex equation-solving contexts where non-bijective operations introduce extraneous solutions requiring careful validation.