Secondary Math · Algebra
Quadratic Equations
All three O-Level methods — factorisation, completing the square, and the quadratic formula — explained with worked examples and exam technique for Singapore Sec 2–4 students.
Which Method Should I Use?
| Method | Use when… | Speed |
|---|---|---|
| Factorisation | Equation has neat integer roots; question says "by factorisation" | Fastest |
| Quadratic Formula | Decimal/surd roots required; no method specified; safety net | Medium |
| Completing the Square | Question explicitly asks for it; finding vertex of parabola; proof questions | Slower |
Factorisation
Use when: Integer or simple fraction roots; equation factors cleanly
- 1Rearrange to ax² + bx + c = 0
- 2Find two numbers that multiply to ac and add to b
- 3Split the middle term and factorise by grouping
- 4Set each factor to zero and solve
Worked Example
Solve x² + 5x + 6 = 0
- Find factors of 6 that add to 5: 2 and 3
- (x + 2)(x + 3) = 0
- x = −2 or x = −3
Quadratic Formula
Use when: Non-integer roots or when factorisation is not obvious
- 1Identify a, b, c from ax² + bx + c = 0
- 2Calculate discriminant: Δ = b² − 4ac
- 3Apply x = (−b ± √Δ) / 2a
- 4Round to required decimal places or leave in surd form
Worked Example
Solve 2x² − 3x − 2 = 0 (correct to 2 d.p.)
- a = 2, b = −3, c = −2
- Δ = 9 + 16 = 25
- x = (3 ± 5) / 4
- x = 2 or x = −0.50
Completing the Square
Use when: Proof questions, finding vertex of parabola, or when formula gives messy surds
- 1Divide by a (if a ≠ 1)
- 2Move constant to the right
- 3Add (b/2a)² to both sides
- 4Write left side as (x + b/2a)²
- 5Square-root both sides and solve
Worked Example
Solve x² + 6x − 7 = 0 by completing the square
- x² + 6x = 7
- x² + 6x + 9 = 7 + 9 = 16
- (x + 3)² = 16
- x + 3 = ±4
- x = 1 or x = −7
The Discriminant: Know Your Roots Before You Solve
Before picking a method, calculate Δ = b² − 4ac
Δ > 0
Two distinct real roots
Δ = 0
One repeated root (tangent to x-axis)
Δ < 0
No real roots (parabola above/below x-axis)
Frequently Asked Questions
What are the three methods for solving quadratic equations in O-Level Math?
The Singapore O-Level syllabus (4048/4049) requires three methods: (1) Factorisation — split the middle term or use the cross-method; (2) Completing the square — rewrite as (x + p)² = q then square-root both sides; (3) Quadratic formula — x = (−b ± √(b²−4ac)) / 2a. The paper will usually specify which method to use; if it does not, factorisation is fastest when the equation factors neatly.
When is a quadratic equation first taught in Singapore secondary school?
Quadratic expressions and equations are introduced in Secondary 2 (G3 track) as part of the Algebra strand. Students learn factorisation and the graphical interpretation of roots. The quadratic formula and completing the square are typically taught in Secondary 3, ahead of the O-Level examination in Secondary 4.
How do I know if a quadratic equation has real roots?
Use the discriminant: Δ = b² − 4ac. If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one repeated real root (the parabola touches the x-axis). If Δ < 0, there are no real roots (the parabola does not cross the x-axis). In O-Level questions, 'show that the equation has no real roots' always means proving Δ < 0.
What are the most common mistakes students make with quadratic equations?
The four most common errors are: (1) Forgetting to set the equation equal to zero before factorising — e.g., treating x² + 3x = 10 as x(x+3) = 10; (2) Sign errors when applying the quadratic formula, especially with negative b; (3) Incomplete factorisation — stopping at 2(x² − 5x + 6) without factorising the bracket; (4) Rounding mid-calculation instead of only at the final step when using the formula.
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