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UAT-UK SPEC: MM1MODULE: MATHEMATICS 2TOPIC ID: ARCH-MM-02

Disguised Radical Equations & Principal Square Root Domain Filtering (√x ≥ 0)

Master Disguised Radical Equations & Principal Square Root Domain Filtering (√x ≥ 0) on the ESAT Mathematics 2 paper. Traditional classroom derivations waste valuable seconds. Using the 2-step Sub-90 invariant elimination method compresses solve time to 40s, saving +110s per question.

Standard Classroom Time
150s
Textbook multi-line derivations bleed exam clock
Sub-90 Target Pace
40s
Comfortably below the 88.8s per-question limit
Time Saved for Review
+110s
Banks clock buffer across 27 questions

2-Step Sub-90 Shortcut Formula

How to eliminate intermediate algebra and solve under 45 seconds.

Launch Interactive Lesson
STEP 1 • SETUP INVARIANT (LINE 1)10–15s
Line 1 (Domain Filter x≥0\sqrt{x} \ge 0): Let u=x≥0u = \sqrt{x} \ge 0; factor u2−1u−6=(u−3)(u+2)=0u^2 - 1u - 6 = (u - 3)(u + 2) = 0.

Establish the structural invariant and eliminate impossible options immediately.

STEP 2 • FAST SOLVE (LINE 2)15–25s
Line 2 (Prune & Square): Reject u=−2<0u = -2 < 0 immediately; only u=3  ⟹  x=32=9u = 3 \implies x = 3^2 = 9 (25s).

Evaluate the simplified expression using integer factoring or unit elimination.

Distractor Trap Forensics

The predictable cognitive traps engineered by examiners into the wrong options.

TRAP #1fence_post_or_domain_trap

Counting extraneous roots where sqrt(f(x)) = g(x) < 0 created by squaring both sides

Violating the UAT-UK convention that √(x) denotes strictly the non-negative root (√(x) >= 0).

Sub-90 Fix: Test invariants before expanding equations.
TRAP #2sign_or_direction_flip

Writing sqrt((a - b*√(c))²) = a - b*√(c) when a < b*√(c)

Failing to take the positive magnitude |a - b*√(c)| = b*√(c) - a.

Sub-90 Fix: Test invariants before expanding equations.
TRAP #3factor_of_two_or_stoichiometry_trap

Counting a repeated root (u - r)² = 0 as 2 distinct x-solutions

Failing to check discriminant b² - 4ac = 0 multiplicity.

Sub-90 Fix: Test invariants before expanding equations.

Try 1 Timed Question: Disguised Radical Equations & Principal Square Root Domain Filtering (√x ≥ 0)

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Full Lesson All 15 Questions (ARCH-MM-02)
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ARCH-MM-02: Disguised Radical Equations & Principal Square Root Domain Filtering (√x ≥ 0) (Sample)Mathematics 2
00:00.0
SUB-90 ZONE (<40S)
0.0s40s Sub-9090s ESAT Budget150s Classroom
Question 1 of 1ARCH-MM-02 (6-Card Lesson)Format: Standard 5-Option
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Question 1 of 1 • 90s Target

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