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UAT-UK SPEC: MM1MODULE: MATHEMATICS 2TOPIC ID: ARCH-MM-15

Quadratic Discriminant (b² - 4ac), Root-Difference Identities & Parameter Range Inequalities

Master Quadratic Discriminant (b² - 4ac), Root-Difference Identities & Parameter Range Inequalities 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 +115s per question.

Standard Classroom Time
155s
Textbook multi-line derivations bleed exam clock
Sub-90 Target Pace
40s
Comfortably below the 88.8s per-question limit
Time Saved for Review
+115s
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 (Root-Difference Identity): Memorise ∣α−β∣=b2−4ac∣a∣|\alpha - \beta| = \frac{\sqrt{b^2 - 4ac}}{|a|} directly.

Establish the structural invariant and eliminate impossible options immediately.

STEP 2 • FAST SOLVE (LINE 2)15–25s
Line 2 (Evaluate): 72−4(2)(3)2=252=52\frac{\sqrt{7^2 - 4(2)(3)}}{2} = \frac{\sqrt{25}}{2} = \frac{5}{2} (20s).

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

Using b² - 4ac >= 0 instead of b² - 4ac > 0 for 'two DISTINCT real roots' (or forgetting a != 0 when a is the x² coefficient)

Including the repeated-root boundary Δ = 0 when strict distinctness is required.

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

Dropping the negative sign on constant term c when the equation is written as ax² + bx = k (c = -k)

Substituting c = +k instead of rearranging to ax² + bx - k = 0 first.

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

Checking Δ > 0 only for 'two positive roots' without also intersecting with sum -b/a > 0 and product c/a > 0

Δ > 0 guarantees real roots, but both roots are positive iff Δ >= 0, -b/a > 0, AND c/a > 0.

Sub-90 Fix: Test invariants before expanding equations.

Try 1 Timed Question: Quadratic Discriminant (b² - 4ac), Root-Difference Identities & Parameter Range Inequalities

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Full Lesson All 15 Questions (ARCH-MM-15)
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ARCH-MM-15: Quadratic Discriminant (b² - 4ac), Root-Difference Identities & Parameter Range Inequalities (Sample)Mathematics 2
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SUB-90 ZONE (<35S)
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Question 1 of 1ARCH-MM-15 (6-Card Lesson)Format: Standard 5-Option
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