ARCH-MM-02: Disguised Radical Equations & Principal Square Root Domain Filtering (√x ≥ 0)

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-02: Why the 150s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~150s (Blows 90s Clock)

Rearranging to

\sqrt{x} = \dots

and squaring both sides gives a quadratic in

x

with roots

x = 9

and

x = 4

. Checking both roots back in the original equation takes ~135s and shows

x = 4

fails.

The Fast ShortcutTarget: 40s (Saves 110s)

**Line 1 (Domain Filter

\sqrt{x} \ge 0

):** Let

u = \sqrt{x} \ge 0

; factor

u^2 - 1u - 6 = (u - 3)(u + 2) = 0

.
Line 2 (Prune & Square): Reject

u = -2 < 0

immediately; only

u = 3 \implies x = 3^2 = 9

(25s).

Where Students Lose Time: In 15 past UAT-UK questions (3.66% of Mathematics 2), candidates lose 110s by violating the uat-uk convention that sqrt(x) denotes strictly the non-negative root (sqrt(x) >= 0). Instead, apply MM1 with Maths 1 skill M2.5.

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