ARCH-M1-07: Algebraic Subject Rearrangement, Rational Fractions & Completing the Square

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-M1-07: Why the 140s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~140s (Blows 90s Clock)

Factor

2

out of the first two terms:

2(x^2 - 6x) + 23 = 2[(x - 3)^2 - 9] + 23 = 2(x - 3)^2 - 18 + 23 = 2(x - 3)^2 + 5

. Hence

r = 5

(~95s).

The Fast ShortcutTarget: 40s (Saves 100s)

**Line 1 (Vertex

x = -\frac{b}{2a} = 3

):**

r

is simply the value of the quadratic at its vertex

x = 3

.
**Line 2 (Evaluate at

x = 3

):**

r = 2(3^2) - 12(3) + 23 = 18 - 36 + 23 = 5

(20s).

Where Students Lose Time: In 31 past UAT-UK questions (13.78% of Mathematics 1), candidates lose 100s by adding instead of subtracting the constant square completion term p*q². Instead, apply M4 with Maths 1 skill M4.2.

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