Quadratic equations in one unknown
How to solve quadratic equations by factorisation, the quadratic formula, completing the square and graphs, and how to use the discriminant and the sum and product of roots. Almost every later algebra topic uses it.
What you will learn
- 1.1 Solve by factorising.
- 1.2 Form a quadratic equation from given (real) roots.
- 1.3 Solve by reading the x-intercepts of y = ax² + bx + c.
- 1.4 Solve by the quadratic formula. Foundation students need not write nonreal roots as a ± bi, or simplify surds such as 2√48.
- 1.5 The discriminant Δ = b² − 4ac and the nature of the roots. If Δ < 0, write "the equation has no real roots".
- 1.6 Word problems.
- NF 1.7 Roots and coefficients: α + β = −b/a and αβ = c/a; form equations from them.
- 1.8 Number systems (appreciation only).
- NF 1.9 Add, subtract, multiply and divide complex numbers a + bi.
You need first: Polynomials and factorisation, Surds
Key terms
| English | 中文 |
|---|---|
| quadratic equation | 一元二次方程 |
| root | 根 |
| factor method | 因式分解法 |
| quadratic formula | 二次公式 |
| discriminant | 判別式 |
| real / nonreal roots | 實根／非實根 |
| equal roots (double root) | 相等的實根（二重根） |
| sum and product of roots | 根之和，根之積 |
| complex number, imaginary unit | 複數，虛數單位 |
| real part, imaginary part | 實部，虛部 |
Standard steps
Choose the method
- Move everything to one side: ax² + bx + c = 0.
- Try to factorise (fastest). Then set each factor to 0.
- If it won't factorise, use the formula: x = [−b ± √(b² − 4ac)] / 2a.
- Check one root by substituting.
Why factorising works: if A × B = 0, then A = 0 or B = 0. This only works with 0 on the right.
Nature of roots
| Δ = b² − 4ac | Roots | Graph of y = ax² + bx + c |
|---|---|---|
| Δ > 0 | two distinct real roots | cuts the x-axis twice |
| Δ = 0 | two equal real roots | touches the x-axis once |
| Δ < 0 | no real roots | doesn't meet the x-axis |
"Has real roots" means Δ ≥ 0.
Roots and coefficients (NF)
For ax² + bx + c = 0 with roots α, β:
- α + β = −b/a, αβ = c/a.
- Useful identities: α² + β² = (α + β)² − 2αβ; 1/α + 1/β = (α + β)/αβ.
- An equation with roots p and q: x² − (p + q)x + pq = 0.
Complex numbers (NF)
- i² = −1.
- Add and multiply like algebra, then replace i² by −1.
- Divide: multiply the top and bottom by the conjugate of the bottom (a + bi → a − bi).
Want to check an answer? Quadratic equation calculator shows every step.
Worked examples
(2x + 1)(x − 3) = 0
x = −1/2 or x = 3x = [4 ± √(16 + 4)] / 2
= [4 ± 2√5] / 2
= 2 ± √5Δ = (k − 2)² − 4(1)(4) = 0
(k − 2)² = 16
k − 2 = ±4
k = 6 or k = −2α + β = 3, αβ = −5
α² + β² = 3² − 2(−5) = 19
sum = (α + 1) + (β + 1) = 5
product = αβ + (α + β) + 1 = −5 + 3 + 1 = −1
equation: x² − 5x − 1 = 0(3 + 2i)(1 − i) = 3 − 3i + 2i − 2i² = 5 − i
(2 + i)(1 + 2i) / [(1 − 2i)(1 + 2i)]
= (2 + 5i + 2i²) / 5
= 5i / 5
= iCommon mistakes
- Factorising when the right-hand side isn't 0: x(x − 3) = 4 does not give x = 4.
- Dividing both sides by x and losing the root x = 0.
- In the formula, −b with negative b: if b = −4, then −b = +4.
- Writing Δ = b² − 4ac with c's sign lost.
- Answering "no solution" instead of "no real roots".
- In α + β, forgetting the minus sign: it is −b/a.
How the DSE tests it
- A(1)/A(2): solve a quadratic, often as one step of a longer question (for example, finding where a graph meets the x-axis).
- A(2): find the range of k from Δ.
- Paper 2: nature of roots; NF roots and coefficients; NF complex number arithmetic.
- Section B: Δ = 0 for a tangent to a circle (F5 circles).
Questions you might not ask in class
Why do I need the formula if the calculator solves quadratics?
The calculator is fine for checking. But many questions have letters (k) instead of numbers, or ask for exact surd answers, and only the formula and Δ handle those.
What is i? Is it real?
It isn't on the number line. Mathematicians invented it so that every quadratic has roots. It follows normal algebra rules plus one rule: i² = −1.
Why is 'no real roots' different from 'no solution'?
There are roots, but they are complex numbers, not real numbers.