ARCH-MM-15: Quadratic Discriminant (b^2 - 4ac), Root-Difference Identities & Parameter Range Inequalities

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-15: Why the 155s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~155s (Blows 90s Clock)

Using the quadratic formula, the roots are

\frac{7 \pm \sqrt{7^2 - 4(2)(3)}}{2(2)} = \frac{7 \pm 5}{4}

, giving

\alpha = 3

and

\beta = \frac{1}{2}

, so

\alpha - \beta = \frac{5}{2}

(~110s).

The Fast ShortcutTarget: 40s (Saves 115s)

Line 1 (Root-Difference Identity): Memorise

|\alpha - \beta| = \frac{\sqrt{b^2 - 4ac}}{|a|}

directly.
Line 2 (Evaluate):

\frac{\sqrt{7^2 - 4(2)(3)}}{2} = \frac{\sqrt{25}}{2} = \frac{5}{2}

(20s).

Where Students Lose Time: In 33 past UAT-UK questions (8.05% of Mathematics 2), candidates lose 115s by including the repeated-root boundary δ = 0 when strict distinctness is required. Instead, apply MM1 with Maths 1 skill M4.6.

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