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§ Expressions & Algebra

Manipulate Expressions

§ Expressions & Algebra

Manipulate Expressions

CCSS.6.EECCSS.7.EECCSS.HSA.REI3 min read

Manipulating expressions involves systematically rearranging algebraic statements using inverse operations and algebraic rules to isolate variables or change their form. This fundamental skill encompasses expanding brackets, factorising polynomials, and rearranging formulae to make different variables the subject. The process requires applying operations equally to both sides of equations whilst maintaining mathematical balance.

§ 01

Why it matters

Expression manipulation forms the backbone of advanced mathematics and appears throughout GCSE and A-level courses. Engineers use these techniques when rearranging formulae like V = IR to find resistance R = V/I in electrical circuits. Economists manipulate profit equations P = R - C to isolate revenue R = P + C when analysing business performance. In Year 8, students expand double brackets like (x + 3)(x + 5) = x² + 8x + 15, whilst Year 9 pupils rearrange formulae such as changing v = u + at to find acceleration a = (v - u)/t. Physics students frequently manipulate kinematic equations, and A-level mathematics requires partial fraction decomposition for integration. The skill appears in real contexts from calculating mortgage payments to determining medication dosages based on body weight.

§ 02

How to solve manipulate expressions

Expanding & Factoring

  • Expand single bracket: a(b + c) = ab + ac.
  • Expand double brackets: (a+b)(c+d) = ac + ad + bc + bd (FOIL).
  • Factorise: find the HCF of all terms and write outside the bracket.
  • Factorise quadratics: find two numbers that multiply to c and add to b.

Example: Expand 3(x + 4) = 3x + 12. Factor 6x + 9 = 3(2x + 3).

§ 03

Worked examples

Beginner§ 01

Make x the subject: x + 5 = 17

Answer: x = 12

  1. Subtract 5 from both sides → x = 17 − 5 — To isolate x, subtract 5 from both sides.
  2. Calculate → x = 12 — 17 − 5 = 12.
Easy§ 02

Make x the subject: 8x = 40

Answer: x = 5

  1. Divide both sides by 8 → x = 408 — To isolate x, divide both sides by the coefficient 8.
  2. Calculate → x = 5 — 40 ÷ 8 = 5.
Medium§ 03

Make x the subject: 2x − 11 = -7

Answer: x = 2

  1. Add 11 to both sides → 2x = 4 — Undo the subtraction by adding 11.
  2. Divide both sides by 2 → x = 2 — 4 ÷ 2 = 2.
§ 04

Common mistakes

  • When expanding 3(x + 4), writing 3x + 4 instead of 3x + 12 by forgetting to multiply the constant term by 3.
  • In rearranging 2x + 5 = 13, subtracting 5 from only one side to get 2x = 13 instead of correctly obtaining 2x = 8.
  • When factorising 6x + 9, writing 2(3x + 9) instead of 3(2x + 3) by not finding the highest common factor correctly.
§ 05

Frequently asked questions

What's the difference between expanding and factorising?
Expanding multiplies out brackets to create longer expressions (3(x + 2) becomes 3x + 6), whilst factorising does the reverse by finding common factors to create brackets (3x + 6 becomes 3(x + 2)). They are inverse operations that undo each other.
How do you check if you've rearranged a formula correctly?
Substitute known values into both the original and rearranged formulae to verify they give the same result. For example, if v = u + at becomes a = (v - u)/t, test with v = 20, u = 5, t = 3 to confirm both give consistent answers.
Why must you perform the same operation on both sides of an equation?
Equations represent balance — both sides are equal. Performing different operations on each side breaks this equality, like adding 5kg to one side of balanced scales but not the other. Mathematical balance requires identical operations on both sides.
What's the FOIL method for expanding double brackets?
FOIL stands for First, Outer, Inner, Last — the order for multiplying terms in (a + b)(c + d). First: ac, Outer: ad, Inner: bc, Last: bd. For (x + 2)(x + 3): x², 3x, 2x, 6, giving x² + 5x + 6.
How do you factorise quadratic expressions like x² + 5x + 6?
Find two numbers that multiply to give the constant term (6) and add to give the coefficient of x (5). Here, 2 × 3 = 6 and 2 + 3 = 5, so x² + 5x + 6 = (x + 2)(x + 3).
§ 06

See also

ExpressionFractionFactorTermEquationCoefficient
§ 06

Where to next?

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