Expressions & Simplifying
Collecting Like Terms
Combine terms with the same variable and power.
Worked Example
Simplify 3x + 2y − x + 5y
Group x terms: 3x − x = 2x
Group y terms: 2y + 5y = 7y
Answer: 2x + 7y
Expanding Brackets
Single bracket: Multiply each term inside by the term outside.
Double brackets: Use FOIL (First, Outer, Inner, Last).
Worked Example
Expand (x + 4)(x − 2)
= x² − 2x + 4x − 8
= x² + 2x − 8
Factorising
Single bracket: Take out the HCF.
Quadratic (a=1): Find two numbers that multiply to c and add to b.
Difference of two squares: a² − b² = (a+b)(a−b)
Worked Example
Factorise x² + 5x + 6
Find two numbers that multiply to 6 and add to 5: 2 and 3
= (x + 2)(x + 3)
Key Facts
- Always check your expansion by re-factorising
- "Factorise completely" means take out ALL common factors
- x² − 25 = (x+5)(x−5) is difference of two squares
Solving Equations
Linear Equations
Use inverse operations to isolate the unknown. Do the same thing to both sides.
Worked Example
Solve 4(x + 3) = 2x + 18
Expand: 4x + 12 = 2x + 18
Subtract 2x: 2x + 12 = 18
Subtract 12: 2x = 6
Divide by 2: x = 3
Simultaneous Equations
Elimination: Add or subtract equations to remove one variable.
Substitution: Rearrange one equation and substitute into the other.
Worked Example
Solve: 2x + y = 7 and x − y = 2
Add equations: 3x = 9, so x = 3
Substitute: 3 − y = 2, so y = 1
Solution: x = 3, y = 1
Exam Tip
Always check your answer by substituting back into the original equation.
Inequalities
Solving Linear Inequalities
Solve exactly like equations, but reverse the inequality sign when multiplying or dividing by a negative number.
Worked Example
Solve −3x + 2 > 11
−3x > 9
Divide by −3 (flip the sign!): x < −3
Common Mistake
Forgetting to flip the inequality when dividing by a negative. −2x > 6 gives x < −3, NOT x > −3!
Sequences
nth Term of a Linear Sequence
Find the common difference (d). The nth term = dn + (first term − d).
Worked Example
Find the nth term of: 7, 11, 15, 19, ...
Common difference d = 4
nth term = 4n + b
When n = 1: 4(1) + b = 7, so b = 3
nth term = 4n + 3
Quadratic Sequences
The second differences are constant. If the second difference is k, then the coefficient of n² is k2.
Key Facts
- Arithmetic sequence: constant first differences
- Quadratic sequence: constant second differences
- Geometric sequence: constant ratio between consecutive terms
Straight-Line Graphs
y = mx + c
m = gradient (steepness). c = y-intercept (where the line crosses the y-axis).
Parallel and Perpendicular Lines
Parallel lines have the same gradient.
Perpendicular lines have gradients that multiply to −1. If one gradient is m, the perpendicular gradient is −1m.
Worked Example
Find the equation of the line perpendicular to y = 2x + 1 passing through (4, 3).
Perpendicular gradient = −12
y = −12x + c
Substitute (4,3): 3 = −2 + c, so c = 5
y = −12x + 5
Quadratics
Solving Quadratic Equations
Three methods: factorising, quadratic formula, or completing the square.
Completing the Square
Rewrite x² + bx as (x + b2)² − (b2)²
Worked Example
Write x² + 6x + 2 in completed square form.
x² + 6x = (x + 3)² − 9
So x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7
Turning point: (−3, −7)
Exam Tip
The discriminant b² − 4ac tells you how many solutions: positive = 2 roots, zero = 1 root, negative = no real roots.
Functions (Higher)
Function Notation
f(x) means "a function of x". To evaluate, substitute the value for x.
Composite function fg(x): Apply g first, then f to the result.
Inverse function f−¹(x): Swap x and y, then rearrange for y.
Worked Example
If f(x) = 2x + 3, find f−¹(x).
Let y = 2x + 3
Swap: x = 2y + 3
Rearrange: y = x − 32
f−¹(x) = x − 32
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