Collect like terms, substitute into expressions and formulae, expand single and double brackets, factorise into single brackets and quadratics, simplify algebraic fractions and change the subject of a formula.

What this dot point is asking

Algebraic manipulation is the core skill of the CCEA Number and Algebra strand and the toolkit every other algebra topic relies on. You must collect like terms, substitute numbers into expressions and formulae, expand single and double brackets, factorise into a single bracket and (at Higher tier) into a quadratic, simplify algebraic fractions, and change the subject of a formula. Weakness here costs marks everywhere, because solving equations, working with graphs and proving results all depend on confident manipulation.

Collecting like terms and substitution

Like terms have exactly the same letters to the same powers. You can add or subtract them by combining the numbers in front (the coefficients): $4x + 7x - 2x = 9x$, and $5a + 3b - 2a + b = 3a + 4b$. The terms $x$ and $x^2$ are not like terms and cannot be combined.

Substitution replaces each letter with a given value and then evaluates using the order of operations. If $a = 3$ and $b = -2$, then $a^2 - 4b = 9 - 4(-2) = 9 + 8 = 17$. Take care with negative values: square them in brackets so the sign is handled correctly.

Expanding brackets

Expanding (multiplying out) removes brackets. A single bracket multiplies every inside term by the factor outside: $3(2x - 5) = 6x - 15$. Watch the signs when the factor is negative: $-2(x - 4) = -2x + 8$.

For double brackets, every term in the first bracket multiplies every term in the second, then like terms are collected. A reliable order is first, outer, inner, last. So $(x + 4)(x - 2) = x^2 - 2x + 4x - 8 = x^2 + 2x - 8$. Squaring a bracket is the same process: $(x + 3)^2 = (x + 3)(x + 3) = x^2 + 6x + 9$, and the middle term is double the product, not zero.

Factorising

Factorising is the reverse of expanding. For a single bracket, take out the highest common factor of every term: $6x^2 + 9x = 3x(2x + 3)$. Always check by expanding back.

At Higher tier you factorise a quadratic $x^2 + bx + c$ into two brackets by finding two numbers that multiply to $c$ and add to $b$. For $x^2 + 7x + 12$, the numbers $3$ and $4$ multiply to $12$ and add to $7$, giving $(x + 3)(x + 4)$. The difference of two squares is a special case: $x^2 - 9 = (x + 3)(x - 3)$.

Algebraic fractions and changing the subject

Algebraic fractions simplify by factorising top and bottom and cancelling common factors, exactly as with numerical fractions. So $\dfrac{x^2 - 9}{x + 3} = \dfrac{(x + 3)(x - 3)}{x + 3} = x - 3$.

Changing the subject rearranges a formula so a different letter stands alone.

Why this matters

Every algebra question, and many in geometry, statistics and applied contexts, rests on accurate manipulation. Expanding and factorising are the gateway to solving quadratics; changing the subject lets you rearrange any formula in science or measures; and simplifying fractions appears in proportion and rates of change. CCEA rewards clear, line-by-line working, so set out each step and keep signs under control.