Simplify and manipulate algebraic expressions: collect like terms, expand single and double brackets, factorise (common factors, quadratics and the difference of two squares), and rearrange formulae to change the subject.

What this dot point is asking

OCR references A1 to A6 cover the core algebra toolkit: collecting like terms, substituting into expressions, expanding single and double brackets, factorising (common factors, quadratics and the difference of two squares), and rearranging formulae to change the subject. Algebra is the language the rest of the course is written in, so this manipulation underlies equations, graphs, geometry proofs and rates of change. The techniques are tested on every paper, including the non-calculator one.

Collecting like terms and substituting

Like terms have identical letter parts, so $3x$ and $5x$ combine to $8x$, but $3x$ and $3x^2$ do not. Simplifying an expression means collecting all like terms: $4a + 2b - a + 3b = 3a + 5b$. Terms with different powers stay separate, so $2x^2 + 3x - x^2 + x = x^2 + 4x$. The index laws also apply to algebraic terms when multiplying and dividing: $x^3 \times x^4 = x^7$, $\dfrac{x^6}{x^2} = x^4$, and $(2x^3)^2 = 4x^6$. Treating the number and letter parts separately keeps these tidy.

Substituting replaces letters with numbers, respecting BIDMAS and signs: if $x = -2$, then $3x^2 - x = 3(4) - (-2) = 12 + 2 = 14$. Substitution into a formula is a common worded-question step, so careful arithmetic with negatives matters; squaring a negative gives a positive, which is the usual trap.

Expanding brackets

Expanding removes brackets by multiplying out.

So $3(2x - 5) = 6x - 15$, and $(x + 4)(x - 2) = x^2 + 2x - 8$. The square $(x - 3)^2$ expands to $x^2 - 6x + 9$, not $x^2 + 9$; the middle term is the trap.

Factorising

Factorising rewrites an expression as a product, the reverse of expanding.

So $6x^2 + 9x = 3x(2x + 3)$ (common factor $3x$), and $x^2 + 7x + 12 = (x + 3)(x + 4)$ (since $3 + 4 = 7$ and $3 \times 4 = 12$). The difference of two squares gives $x^2 - 25 = (x + 5)(x - 5)$. At Higher tier you also factorise $ax^2 + bx + c$ by splitting the middle term, and simplify algebraic fractions by cancelling common brackets. For example, $\dfrac{x^2 - 9}{x^2 + 7x + 12}$ factorises to $\dfrac{(x + 3)(x - 3)}{(x + 3)(x + 4)}$, and cancelling the common $(x + 3)$ leaves $\dfrac{x - 3}{x + 4}$. You can only cancel whole factors (brackets), never individual terms, which is a common source of error.

Changing the subject

Rearranging a formula isolates a chosen letter.

The same inverse-operation logic handles squares and roots: to make $r$ the subject of $A = \pi r^2$, divide by $\pi$ then take the square root of the whole right-hand side. When the subject appears twice, factorise it out into a single bracket first, a Higher-tier skill.

Why manipulation underpins everything

Every later topic uses these moves: solving an equation needs expanding and collecting, finding a line's equation needs rearranging into $y = mx + c$, and circle and area problems need factorising. OCR's AO1 marks reward fluent technique, while AO2 rewards setting out each manipulation clearly so a marker can follow the logic even if the final answer slips.