This mini-lesson covers the Algebra strand of OCR GCSE Maths (J560): expanding & factorising, linear & quadratic equations, simultaneous equations, rearranging formulae, sequences (nth term), straight-line graphs and inequalities — with the Higher-tier quadratic-formula work along the way.
Work through each screen, answer the questions as you go (most ask you to type a single number) and collect ⭐ stars. Press Start when you're ready.
To expand a bracket, multiply every term inside by the term outside. Then collect like terms (add together terms with the same letter and power).
Expand and simplify 3(x + 4) + 2(x − 1).
3(x + 4) = 3x + 12 and 2(x − 1) = 2x − 2
3x + 12 + 2x − 2 → collect: (3x + 2x) + (12 − 2) = 5x + 10
Common slip: only like terms combine. 5x + 10 cannot be simplified to 15x — the 10 has no x, so it stays separate.
Factorising puts a bracket back in. Take out the highest common factor of every term (a number and/or a letter):
Factorise 6x + 9.
HCF of 6 and 9 is 3, so pull out 3: 3(2x + 3)
Check by expanding: 3 × 2x = 6x, 3 × 3 = 9. ✓
Factorise x² + 5x + 6.
Find two numbers that multiply to 6 and add to 5: that's 2 and 3.
So x² + 5x + 6 = (x + 2)(x + 3)
Difference of two squares: x² − 9 = x² − 3² = (x − 3)(x + 3). No middle term because +3x and −3x cancel.
An equation has an equals sign. To solve, do the same operation to both sides until the letter is on its own. Undo operations in reverse order.
Solve 4x + 7 = 23.
Subtract 7 from both sides: 4x = 16
Divide both sides by 4: x = 4
Solve 5x − 3 = 2x + 12.
Subtract 2x from both sides: 3x − 3 = 12
Add 3: 3x = 15, then ÷ 3: x = 5
Golden rule: whatever you do to one side, do to the other. Move the smaller x-term across first to keep the x-coefficient positive.
Changing the subject of a formula uses the same "do it to both sides" idea as solving an equation — but you keep letters instead of numbers.
Make x the subject of y = 2x + 1.
Subtract 1: y − 1 = 2x
Divide by 2: x = (y − 1) ÷ 2
Watch out: you must divide the whole side by 2, not just the y. Writing x = y − 1 ÷ 2 loses the bracket and is wrong.
A quadratic contains an x² term. To solve one that factorises, get it equal to 0, factorise, then set each bracket to 0.
Solve x² − 5x + 6 = 0.
Two numbers that multiply to 6 and add to −5: that's −2 and −3.
(x − 2)(x − 3) = 0, so x − 2 = 0 or x − 3 = 0
x = 2 or x = 3 (two solutions)
Key idea: if two things multiply to zero, at least one must be zero. That's why each bracket gives a solution.
When a quadratic won't factorise nicely, use the formula. For ax² + bx + c = 0:
Solve x² − 4x − 5 = 0 using the formula (a = 1, b = −4, c = −5).
b² − 4ac = 16 − (4 × 1 × −5) = 16 + 20 = 36, and √36 = 6.
x = (4 ± 6) ÷ 2 → x = 5 or x = −1
Higher only: solving quadratics with the formula (and by completing the square) is on the Higher tier of J560. Watch the signs — a negative b becomes −(−4) = +4.
A linear (arithmetic) sequence goes up (or down) by the same amount each time — the common difference. The nth term rule is:
For 3, 7, 11, 15, …: d = 4 and a = 3, so nth term = 4n + (3 − 4) = 4n − 1.
Check: n = 1 gives 4 − 1 = 3. ✓
Two equations, two unknowns. Add or subtract the equations to eliminate one letter, solve for the other, then substitute back.
Solve x + y = 10 and x − y = 4.
Add the equations: (x + x) + (y − y) = 10 + 4 → 2x = 14
So x = 7. Substitute into x + y = 10: 7 + y = 10 → y = 3
Why add here? The +y and −y cancel, leaving just x. If the signs matched (both +y), you would subtract to eliminate instead.
Every straight line can be written as y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where it crosses the y-axis).
Read it off directly: in y = 2x + 1, the number in front of x (2) is the gradient and the lone number (+1) is the y-intercept.
An inequality uses < ≤ > ≥ instead of =. Solve it just like an equation — with one extra rule.
Solve 2x + 1 < 9.
Subtract 1: 2x < 8
Divide by 2: x < 4
The one extra rule: if you multiply or divide by a negative number, you must flip the inequality sign. E.g. −x < 3 becomes x > −3.
Tap an expression on the left, then its matching factorised form on the right.
A linear expression has x to the power 1 (no x²); a quadratic has an x² term as its highest power. Tap an expression, then tap the box it belongs in.
Expand: multiply out brackets, then collect like terms.
Factorise: the reverse — take out a common factor, or use two numbers that multiply/add correctly for a quadratic; spot the difference of two squares.
Linear equations: do the same to both sides until x is alone.
Rearranging: same idea, changing the subject of a formula.
Quadratics: set to 0, factorise, each bracket gives a root; use the formula when it won't factorise (Higher).
Sequences: nth term = dn + (a − d) for a linear sequence.
Simultaneous: add or subtract to eliminate a letter.
Graphs: y = mx + c — m is the gradient, c the y-intercept.
Inequalities: solve like equations, but flip the sign when ×/÷ by a negative.
You've covered the Algebra strand of OCR GCSE Maths (J560) — Foundation content plus the Higher-tier quadratic formula. Press Finish to see your score.
You've worked through Algebra for OCR GCSE Maths (J560). 🎉
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Next: test yourself in the Evaluate stage Confidence Quiz, then lock it in with Verify.