Quadratic Equations in One Unknown
Master the discriminant, root transformation identities, and quadratic inequalities. Learn to spot the subtle wording traps that examiners use to filter candidates.
The Core Concept
The Anatomy of a Quadratic: A quadratic equation is represented as
ax² + bx + c = 0 (where a ≠ 0). Geometrically, solving this equation is the same as finding where the parabola y = ax² + bx + c crosses the x-axis (y = 0).DSE Quadratic Equations in One Unknown Cheat Sheet — 4 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. The Discriminant | Δ = b² − 4ac• Δ > 0: 2 distinct real roots • Δ = 0: 1 repeated real root • Δ < 0: No real roots | Check your inequality signs: • "Real roots" ⟹ Δ ≥ 0• "No real roots" ⟹ Δ < 0 |
| 2. Sum & Product of Roots | For ax² + bx + c = 0 with roots α, β:• Sum (S): α + β = −b / a• Product (P): αβ = c / a | Express transformed expressions in terms of S and P: • α² + β² = (α + β)² − 2αβ• (α − β)² = (α + β)² − 4αβ |
| 3. Forming New Equations | To form a quadratic equation with roots α′ and β′: 1. Calculate new Sum: S′ = α′ + β′2. Calculate new Product: P′ = α′β′3. Equation: x² − S′x + P′ = 0 | For symmetric transformations (e.g., 2α, 2β or α + 1, β + 1), substitute back to check your algebra. |
| 4. Quadratic Inequalities | For x² + bx + c ≥ 0 or ≤ 0 with roots r₁ < r₂:• If ≥ 0: Outside range (x ≤ r₁ or x ≥ r₂)• If ≤ 0: Inside range (r₁ ≤ x ≤ r₂) | Always make the x² coefficient positive before applying the "inside/outside" shortcut. |
💡 Exam Traps & Secrets
⚠️ The "Real Roots" TrapDSE questions often state: "The quadratic equation has real roots." Many students write
Δ > 0. This is incorrect and will cost you points. "Real roots" includes the possibility of a repeated root, so the correct condition is: Δ ≥ 0.⚠️ The Root Identity Cheat SheetMemorize these algebraic transformations to save time in Paper 1 Section A:
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α² + β² = (α + β)² − 2αβ•
(α − β)² = (α + β)² − 4αβ•
1/α + 1/β = (α + β) / αβ•
α³ + β³ = (α + β)[(α + β)² − 3αβ]⚠️ Negative Quadratic Inequality TrapIf you need to solve
−x² + 3x + 10 ≥ 0, do not apply the "outside range" shortcut immediately. You must multiply the entire inequality by −1 and flip the inequality sign first: x² − 3x − 10 ≤ 0. Now, factorize to find roots −2 and 5. Because the sign is now ≤ 0, the answer is "inside": −2 ≤ x ≤ 5.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Discriminant inequalities)Find the range of values of k such that the equation
3x² − kx + 12 = 0 has two distinct real roots.(Sum/product properties and root transformations)The roots of
x² − 5x + 3 = 0 are α and β. Find the value of α² + β², and form a new quadratic equation whose roots are 2α and 2β.(Graphical inequalities & factorizing quadratic trinomials)Solve the quadratic inequality
x² + 3x − 10 ≥ 0.Quadratic Equations in One Unknown Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
Quadratic Equations and Complex Numbers · Form 4
20 MC questions · 7 long questions
Frequently Asked Questions
When Δ = b² − 4ac > 0. For a repeated root, Δ = 0. For no real roots, Δ < 0. Always use the strict inequality Δ > 0 when the question says "two distinct real roots". If the question only says "real roots" (without "distinct"), the correct condition is Δ ≥ 0.
New sum = 2α + 2β = 2(α + β). New product = (2α)(2β) = 4αβ. The equation is x² − (new sum)x + (new product) = 0. Always express the new sum and product in terms of the original sum (−b/a) and product (c/a) before simplifying.
Paper 1 Section A tests discriminant conditions and root expressions (α²+β², 1/α+1/β, etc.). Paper 2 multiple-choice tests the graphical interpretation — how many times does y = ax² + bx + c meet the x-axis. For inequalities, remember to make the x² coefficient positive before applying the inside/outside shortcut.