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

The Eduqas algebra content opens with manipulation: the toolkit of simplifying, expanding, factorising and rearranging that every later topic depends on. You must collect like terms, expand single and double brackets, factorise by taking out common factors and by reversing a double expansion (including the difference of two squares), and change the subject of a formula. Manipulation appears on both components and at both tiers, and weak manipulation leaks marks across geometry, ratio and the rest of algebra, so fluency here is the single highest-value investment in the course.

Simplifying and the index laws

Collecting like terms is adding the coefficients of terms with identical letters and powers: $4a + 7b - a + 2b = 3a + 9b$. Only like terms combine, so $3x$ and $3x^2$ stay separate. When you multiply algebraic terms, multiply the numbers and add the indices.

Expanding brackets

To expand a single bracket, multiply each inside term by the term outside: $3(2x - 5) = 6x - 15$. For two brackets, multiply every term in the first by every term in the second. FOIL (First, Outer, Inner, Last) is a way to track the four products.

So $(x + 4)(x - 2) = x^2 - 2x + 4x - 8 = x^2 + 2x - 8$. Squaring a bracket is a double expansion, not just squaring each term: $(x + 3)^2 = (x+3)(x+3) = x^2 + 6x + 9$, where the $6x$ middle term is the one most often dropped.

Factorising

Factorising is the reverse of expanding. Always look first for a common factor in every term.

For $x^2 + 7x + 12$, two numbers multiply to $12$ and add to $7$: those are $3$ and $4$, so it factorises as $(x + 3)(x + 4)$. A special case is the difference of two squares, which has no middle term: $x^2 - 25 = (x + 5)(x - 5)$ and $9x^2 - 16 = (3x + 4)(3x - 4)$.

Changing the subject

Rearranging a formula uses the same balancing as solving an equation: do the same operation to both sides, undoing operations in reverse order.

When the new subject appears more than once, gather its terms on one side, factorise it out, then divide. For example to make $x$ the subject of $ax = bx + c$, write $ax - bx = c$, factorise to $x(a - b) = c$, and divide to get $x = \dfrac{c}{a - b}$. This factorise-then-divide pattern is the Higher-tier twist examiners use to separate strong candidates, so practise spotting when the subject is hidden inside two or more terms.

Why manipulation matters

Every algebraic technique downstream depends on this one. Solving equations needs expanding and rearranging; the quadratic formula needs a quadratic in standard form; straight-line and proportion work needs the subject changed; even geometry proofs are written in manipulated algebra. Because Eduqas weights reasoning (AO2) and problem solving (AO3) at half the marks, questions often hide the manipulation inside a context, so recognising when to factorise or rearrange is as important as the mechanics.