Algebra
Expressions, equations, inequalities, sequences, graphs, and functions.
In this topic
Algebraic Expressions
An algebraic expression uses letters to represent unknown values. Terms are the parts of an expression separated by + or -.
Like terms have the same letter(s) raised to the same power. You can only add or subtract like terms.
Simplifying: 3x + 5y - 2x + y = x + 6y
Key Points
- Like terms: same letter, same power — can be collected
- Multiplying: 3a × 2b = 6ab
- Expanding brackets: multiply every term inside by the term outside
- Factorising: the reverse of expanding — take out common factors
Example Questions
2Expand and simplify: 3(2x + 4) - 2(x - 3)
6x + 12 - 2x + 6 = 4x + 18
[2 marks]
2Factorise completely: 6x²y + 9xy²
3xy(2x + 3y)
[2 marks]
Expanding Brackets
Single brackets: multiply each term inside by the term outside. 3(x + 4) = 3x + 12
Double brackets (FOIL method): (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
Triple brackets: expand two first, then expand the result with the third.
Key Points
- Single bracket: multiply each term
- Double bracket: First, Outside, Inside, Last (FOIL)
- Difference of two squares: (a+b)(a-b) = a² - b²
- Perfect square: (a+b)² = a² + 2ab + b²
Example Questions
2Expand (2x - 3)(x + 5)
2x² + 10x - 3x - 15 = 2x² + 7x - 15
[2 marks]
2Expand and simplify (x + 4)²
x² + 8x + 16
[2 marks]
Factorising
Factorising is the reverse of expanding — putting an expression back into brackets.
Types of factorising: 1. Common factor: 6x + 9 = 3(2x + 3) 2. Quadratic (x² + bx + c): find two numbers that multiply to c and add to b 3. Difference of two squares: a² - b² = (a+b)(a-b) 4. Quadratic (ax² + bx + c where a ≠ 1): use grouping method
Key Points
- Always look for a common factor first
- For x² + bx + c: find two numbers that multiply to c, add to b
- Check by expanding your answer
Example Questions
2Factorise x² + 7x + 12
(x + 3)(x + 4) — since 3 × 4 = 12 and 3 + 4 = 7
[2 marks]
1Factorise 25x² - 16
(5x + 4)(5x - 4) — difference of two squares
[1 mark]
Solving Equations
An equation has an equals sign and can be solved to find the value of the unknown.
To solve an equation, do the same operation to both sides to isolate the variable.
Linear equations: ax + b = c → solve by inverse operations Equations with unknowns on both sides: collect terms on one side
Key Points
- Whatever you do to one side, do to the other
- Undo operations in reverse order (reverse BIDMAS)
- With fractions: multiply both sides by the denominator
- Always check your solution by substituting back
Example Questions
2Solve: 5x - 3 = 2x + 12
3x = 15, so x = 5
[2 marks]
2Solve: (x + 3)/4 = 5
x + 3 = 20, so x = 17
[2 marks]
Solving Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0. Three methods to solve:
1. Factorising: x² + 5x + 6 = 0 → (x+2)(x+3) = 0 → x = -2 or x = -3
2. Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
3. Completing the square: x² + bx + c → (x + b/2)² - (b/2)² + c
The discriminant b² - 4ac tells you about the roots: - Positive: two real solutions - Zero: one repeated solution - Negative: no real solutions
Key Points
- Always try factorising first — it's quickest
- Quadratic formula works for ALL quadratics
- If it can't factorise, use the formula
- A quadratic can have 0, 1, or 2 solutions
Example Questions
2Solve x² - 5x + 6 = 0
(x-2)(x-3) = 0, so x = 2 or x = 3
[2 marks]
3Solve 2x² + 3x - 5 = 0 using the quadratic formula.
a=2, b=3, c=-5. x = (-3 ± √(9+40))/4 = (-3 ± 7)/4. x = 1 or x = -2.5
[3 marks]
Simultaneous Equations
Simultaneous equations are two (or more) equations with two (or more) unknowns, solved together.
Elimination method: Make the coefficients of one variable the same, then add or subtract. Substitution method: Rearrange one equation to express one variable in terms of the other, then substitute.
Key Points
- Elimination: match coefficients, then add/subtract
- Substitution: works well when one equation is already rearranged
- Always find BOTH unknowns
- Check by substituting both values into the other equation
Example Questions
3Solve: 2x + 3y = 13 and x - y = 1
From equation 2: x = y + 1. Substitute: 2(y+1) + 3y = 13 → 5y + 2 = 13 → y = 2.2. Hmm, let me redo: 2y + 2 + 3y = 13 → 5y = 11 → y = 11/5. Actually, let's use elimination: multiply eq2 by 3: 3x - 3y = 3. Add to eq1: 5x = 16, x = 3.2, y = 2.2
[3 marks]
Sequences
A sequence is an ordered list of numbers following a pattern. The nth term formula gives you any term without listing them all.
Arithmetic (linear) sequences have a constant difference between consecutive terms. nth term = dn + (a - d), where d = common difference, a = first term.
Geometric sequences multiply by a constant ratio each time.
Key Points
- Find the common difference for arithmetic sequences
- nth term of arithmetic: a + (n-1)d
- Geometric: multiply by common ratio r each time
- Quadratic sequences have a changing difference — find the second difference
Example Questions
2Find the nth term of the sequence 5, 8, 11, 14, ...
Common difference = 3. nth term = 3n + 2
[2 marks]
2A geometric sequence starts 2, 6, 18, 54, ... Find the 8th term.
Common ratio = 3. 8th term = 2 × 3⁷ = 2 × 2187 = 4374
[2 marks]
Graphs
Graphs provide a visual representation of mathematical relationships. The most common types at GCSE are:
Linear graphs: y = mx + c (straight lines), where m is the gradient and c is the y-intercept. Quadratic graphs: y = ax² + bx + c (parabolas/U-shapes). Cubic, reciprocal, and exponential graphs each have distinctive shapes.
Key Points
- Gradient = rise/run = change in y / change in x
- Parallel lines have the same gradient
- Perpendicular lines: gradients multiply to give -1
- The turning point of a quadratic is its minimum or maximum
Example Questions
3Find the equation of the line passing through (1, 3) and (3, 7).
Gradient = (7-3)/(3-1) = 2. y = 2x + c. Using (1,3): 3 = 2(1) + c, c = 1. Equation: y = 2x + 1
[3 marks]
3A line is perpendicular to y = 3x + 1 and passes through (6, 2). Find its equation.
Perpendicular gradient = -1/3. y = -1/3 x + c. 2 = -1/3(6) + c → c = 4. y = -1/3 x + 4
[3 marks]
Inequalities
Inequalities use < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
Solve inequalities like equations, but remember: if you multiply or divide by a negative number, FLIP the inequality sign.
On a number line: - Open circle ○ means < or > (not included) - Closed circle ● means ≤ or ≥ (included)
Key Points
- Solve like equations, but flip the sign when multiplying/dividing by a negative
- Represent solutions on a number line
- Double inequalities: solve the middle part, applying to all three sections
Example Questions
2Solve 3x + 5 < 20
3x < 15, so x < 5
[2 marks]
3List the integer values of n where -3 < 2n + 1 ≤ 7
-4 < 2n ≤ 6 → -2 < n ≤ 3. Integers: -1, 0, 1, 2, 3
[3 marks]