Think of factorising as reverse-expanding. Normally you'd multiply out brackets like (x − 2)(x + 5) to get x² + 3x − 10. Factorising does the opposite: you're given x² + 3x − 10 = 0, and your job is to find the two brackets that would multiply together to make it.
Why bother? Because once it's in the form (something)(something) = 0, you can use a genuinely powerful trick:
The Zero Product Rule
If two numbers multiply together to give zero, one of them must be zero. There's no other way to get zero from a multiplication (other than 0 × 0). So if (x − 3)(x + 7) = 0, then either x − 3 = 0 or x + 7 = 0 — and you solve each bracket separately as its own mini equation.
A The Basic Method
- Rearrange the equation into the form ax² + bx + c = 0 — get zero onto one side. It's easier if you arrange things so that a (the coefficient of x²) is positive.
- Factorise the left-hand side into two brackets.
- Set each bracket equal to zero separately, and solve each one as its own simple equation.
- Write both solutions together using "or" — a quadratic almost always has two answers.
Worked Example
Solve (x − 3)(x + 7) = 0
x − 3 = 0 → x = 3 (add 3 to both sides)
x + 7 = 0 → x = −7 (subtract 7 from both sides)
Two solutions: x = 3 or x = −7
Spot the pattern
When the coefficient of x² is just 1 (like in the example above), the two solutions are simply the numbers in the brackets with the opposite sign. (x − 3) gave x = +3. (x + 7) gave x = −7. This is a handy sanity check, but it only works when a = 1.
B When There Are Numbers in Front of the x's
The process is exactly the same — but you can no longer just flip the sign to get the answer. There's real work to do in each bracket, because now the bracket doesn't just say "x + something", it says "(a number) × x + something".
Worked Example
Solve (2x − 3)(3x + 5) = 0
2x − 3 = 0 → 2x = 3 → x = 3/2 (add 3, then divide by 2)
3x + 5 = 0 → 3x = −5 → x = −5/3 (subtract 5, then divide by 3)
Two solutions: x = 3/2 or x = −5/3
C When x Itself Is a Factor
Sometimes there's no constant term at all — the equation looks like x(x − 4) = 0. It can help to mentally rewrite the lone "x" as "(x − 0)" or just "(x)" so you treat it exactly like any other bracket.
Worked Example
Solve x(x − 4) = 0
(x) = 0 → x = 0
x − 4 = 0 → x = 4
Two solutions: x = 0 or x = 4
The #1 Trap Here
It is extremely tempting to divide both sides of x(x − 4) = 0 by x, leaving x − 4 = 0, and conclude x = 4 is the only answer. Don't do this. Dividing by x silently deletes the x = 0 solution — you're not allowed to divide by an unknown that could be zero. Always factorise and use both brackets.
Practice
Solve x² + 3x − 10 = 0 by factorising.
Practice
Solve 5x² − x = 0 by factorising.
Not every quadratic factorises nicely. Sometimes the numbers just don't split into neat whole-number brackets. That's where the quadratic formula saves you — it works on every single quadratic equation, no matter how ugly the numbers are.
In plain English: pull out the three numbers a, b, and c from your equation, feed them into this formula, and out pop your two solutions — one from using "+" and one from using "−" in the ± symbol.
A How to Use It
- Make sure the equation is arranged as "= 0" on the right-hand side. If it isn't, rearrange it first.
- Read off the values of a, b, and c — including their signs.
- Substitute them into the formula. Put brackets around any negative numbers you substitute — this prevents sign errors, which are the #1 cause of wrong answers here.
- Simplify by hand, or type it into a calculator (once using + for ±, then again using − for ±).
Worked Example
Solve 2x² − 8x − 3 = 0 using the quadratic formula
a = 2, b = −8, c = −3
x = ( −(−8) ± √((−8)² − 4×2×(−3)) ) / (2×2)
Solutions: x = 4.3452... or x = −0.3452...
To 3 s.f.: x = 4.35 or x = −0.345
Examiner Tip
Always check exactly how the question wants your final answer left — "correct to 2 decimal places", "3 significant figures", or "exact (surd) form" all mean something different, and marks are lost for the wrong format even if your method was perfect.
B Giving Answers in Exact (Surd) Form
On non-calculator papers, you can't just type the formula in and read off a decimal — you need to simplify the square root by hand and leave your answer in "surd" form (with a √ symbol in it, kept exact rather than rounded).
Worked Example (continued from above)
(−8)² − 4×2×(−3) = 64 + 24 = 88
√88 = √(4 × 22) = 2√22 (pull out the perfect square factor)
x = (8 ± 2√22) / 4 = (4 ± √22) / 2 (simplify by dividing every term by 2)
x = (4 + √22)/2 or x = (4 − √22)/2
C The Discriminant — Predicting How Many Solutions
The expression under the square root, b² − 4ac, is called the discriminant. You don't even need to finish solving the equation to know how many solutions it has — just calculate this one value first.
Why does this make sense?
You can't square-root a negative number and get a real answer — that's why b² − 4ac < 0 gives no solutions. If b² − 4ac is exactly 0, then ±√0 is just 0, so the "+" and "−" versions of the formula give the same answer — hence only one solution. And bonus fact: if b² − 4ac turns out to be a perfect square (1, 4, 9, 16, 25...), the quadratic could actually have been factorised neatly all along!
Practice
Use the quadratic formula to solve 3x² − 2x − 4 = 0. Give each solution as an exact value in its simplest form.
Practice
Without fully solving, use the discriminant to state how many solutions the equation x² + 4x + 4 = 0 has.
On calculator papers, you're expected to use your graphic display calculator's built-in polynomial solver rather than grinding through the formula by hand — but you still need to show your working clearly, because marks are awarded for method, not just the final answer.
- Open the menu and find the option for solving a polynomial (sometimes inside an "equation solver" section, depending on your model).
- Select the degree/order = 2 — this tells the calculator it's a quadratic.
- Type in the coefficients a, b, and c from your equation — first rearranging it into the form ax² + bx + c = 0 if it isn't already.
- Press solve to view both solutions.
Exact vs. Decimal
Good calculators present answers in exact form first (like −3 + √13) rather than a rounded decimal, and let you switch between the two views. If there's more working to do afterwards, keep the exact form. If the solutions are your final answer, check what the question is actually asking for — an exact answer, or a decimal rounded to a stated number of significant figures.
What to Still Write Down
Even though you don't need to show the quadratic formula step-by-step on a calculator paper, you should still write: (1) the equation you typed into the calculator, including any rearranging you did first, and (2) both solutions the calculator gave you, in exact form where possible. This protects your method marks if a typo creeps in.
Worked Example
Use your calculator to solve 10x = 19 − 6x², giving solutions to 4 s.f.
Rearrange: 6x² + 10x − 19 = 0
Enter coefficients 6, 10, −19 into the polynomial solver
Exact form: x = (−5 + √139)/6 or x = (−5 − √139)/6
To 4 s.f.: x = 1.132 or x = −2.798
Practice
Rearrange 4x² = 12 − 5x into the form ax² + bx + c = 0, ready to type into a calculator's polynomial solver.
Quadratic equation
Any equation that can be written as ax² + bx + c = 0, where a ≠ 0. The x² term is what makes it "quadratic" rather than linear.
Zero product rule
If (bracket 1) × (bracket 2) = 0, then bracket 1 = 0 OR bracket 2 = 0. This is the entire justification for solving by factorising.
The quadratic formula
x = ( −b ± √(b² − 4ac) ) / 2a — works for every quadratic, factorisable or not. Always put brackets around negative values of a, b, c when substituting.
Discriminant
b² − 4ac. Positive → 2 solutions. Zero → 1 (repeated) solution. Negative → 0 real solutions.
Surd form
Leaving a square root un-rounded (e.g. 2√22 rather than 9.38...), simplified using surd rules — pull out the largest perfect square factor from underneath the root.
Dividing by xIn x(x − 4) = 0, dividing both sides by x loses the x = 0 solution. Always keep both brackets and solve each one.
Sign errors with negativesWhen b or c is negative, always substitute it in brackets, e.g. −(−8) and (−8)². Skipping the brackets is the #1 cause of wrong formula answers.
Forgetting "= 0" firstThe quadratic formula and factorising both require the equation to equal zero on one side before you start — rearrange first if it doesn't.
Not simplifying surds fully√88 left as √88 is not fully simplified — always pull out the largest perfect square factor, e.g. √88 = 2√22.
Wrong answer formatGiving a decimal when the question asks for an exact (surd) answer, or vice versa, loses marks even with correct method — always re-read what's being asked.
Skipping working on calculator papersEven when your calculator solves it instantly, write down the equation you entered and both solutions — this protects method marks if there's a typing error.