Summary — What's in this Chapter
- Solving by factorising — split the quadratic into two brackets and set each to zero.
- The quadratic formula — a "works every time" formula, especially good with large or ugly numbers.
- Completing the square — rewrites the quadratic so x appears only once; reveals the turning point.
- Choosing the right method — a decision-making skill examiners specifically test.
- Hidden quadratics — equations in disguise (x⁴, √x, 2ˣ, tan x) that are secretly quadratic in structure.
- The discriminant (b² − 4ac) — tells you how many real roots a quadratic has, without solving it.
- Quadratic graphs — the parabola shape, roots, y-intercept, and turning point.
- Quadratic inequalities — solving ax² + bx + c > 0 (or <, ≤, ≥) using a sketch.
- Inequalities on graphs — shading a region that satisfies several inequalities (linear + quadratic) at once.
1. Solving Quadratics by Factorising
The Core Idea
Think of a quadratic equation like (x − 3)(x + 7) = 0 as two numbers being multiplied together to make zero. The only way two numbers multiply to give zero is if at least one of them IS zero. There's no other way — you can't multiply two non-zero numbers and land on zero. So if A × B = 0, then either A = 0 or B = 0 (or both).
This is the entire trick behind factorising. Once your quadratic is written as two brackets multiplied together and set equal to zero, you just solve each bracket separately.
ax² + bx + c = 0 → factorise → (bracket 1)(bracket 2) = 0
Rearrange so zero is on one side (ideally with a positive x² term) before you factorise.
Worked Example — Step by Step
Solve (2x − 3)(3x + 5) = 0
Step 1 — Set the first bracket to zero: 2x − 3 = 0 → 2x = 3 → x = 3/2
Step 2 — Set the second bracket to zero: 3x + 5 = 0 → 3x = −5 → x = −5/3
Step 3 — Write both solutions together: x = 3/2 or x = −5/3
⚠️ The classic mistake — don't lose a solution!
To solve x(x − 4) = 0, it's tempting to divide both sides by x, leaving x − 4 = 0, giving only x = 4. But this silently deletes the x = 0 solution. Dividing by an unknown that could be zero is illegal in algebra — always treat x as its own bracket instead. Correct answer: x = 0 or x = 4.
Practice Question 1
Solve x(5x − 1) = 0.
Practice Question 2
Solve (8x + 7)(2x − 3) = 0.
2. The Quadratic Formula
Why It Always Works
Not every quadratic factorises nicely — sometimes the numbers are just too ugly or "random". The quadratic formula is the universal backup plan: plug in a, b and c from ax² + bx + c = 0, and it spits out both solutions, guaranteed, every single time (as long as real solutions exist).
x = ( −b ± √(b² − 4ac) ) / 2a
In plain words: take negative b, plus-or-minus the square root of "b squared minus 4ac", all divided by 2a.
🛡️ Safety habit: bracket your negatives
Worked Example — Step by Step
Solve 2x² − 5x + 2 = 0 using the formula
Step 1 — Identify: a = 2, b = −5, c = 2
3. Completing the Square
The Core Idea