Substitution & Using Formulae, — AQA GCSE Mathematics

SPEC A2, A5 • ALGEBRA

Substitution & Using Formulae

Replace letters with values, put negatives in brackets, apply BIDMAS — the rule that turns algebra into arithmetic.

What is Substitution?

Substitution is where algebra meets arithmetic. You take an expression or formula full of letters and swap each letter for a given number.

Substitution

Substitution means replacing every letter in an expression or formula with its given numerical value, then evaluating the result using BIDMAS.

Where You'll See It

Expressions: evaluate 3a² − 2b + ab for given a and b

Standard formulae: find the area of a trapezium from its given dimensions

Scientific formulae: density, speed, kinematics — often unfamiliar and given in the question

Functions: find f(−2) when f(x) = 3x² − 5x + 2

The Three Rules of Substitution

Every substitution question in GCSE Maths obeys the same three rules. Get these right and you will not drop a mark.

Rule 1 — Replace Every Letter

Wherever a letter appears, swap it for its value. Every single time — don't miss one.

Rule 2 — Wrap Negatives in Brackets

If a value is negative, always put it in brackets when you substitute. This is the single most common place students lose marks in this topic.

Rule 3 — Follow BIDMAS

Substitution doesn't change the order of operations. Brackets first, then Indices, then Division/Multiplication, then Addition/Subtraction.

Worked Example: Expression

01Evaluate an Expression [2 marks]

Evaluate 3a² − 2b + ab when a = −2 and b = 5.

Step 1 — Replace every letter. Brackets around each negative.

= 3(−2)² − 2(5) + (−2)(5)

Step 2 — Indices first (BIDMAS): (−2)² = 4.

= 3(4) − 2(5) + (−2)(5)

Step 3 — Carry out all multiplications.

= 12 − 10 + (−10)

Step 4 — Simplify: adding a negative is the same as subtracting.

= 12 − 10 − 10

Step 5 — Add and subtract left to right.

= −8

The Negative Brackets Trap

This is the single trap that catches the most students on Paper 1. Writing brackets around a negative — or not writing them — can flip the sign of your whole answer.

Examiner Alert

Students write: Substituting x = −3 into x² gives −9.

Why it loses marks: (−3)² = 9 — a negative squared is positive.

What to write instead: always wrap the negative in brackets: (−3)² = 9, never −9.

BIDMAS Is Not Optional

Substitution isn't permission to forget the order of operations. The whole expression must be evaluated using BIDMAS, even after you've put the numbers in.

BIDMAS Order

Brackets → Indices → Division/Multiplication → Addition/Subtraction

Example: Evaluate 2 + 3x² when x = 4.

Wrong: do the addition first.

2 + 3(4)² → (2 + 3)(4)² = 5 × 16 = 80 ✗

Right: indices, then multiplication, then addition.

2 + 3(4)² = 2 + 3(16) = 2 + 48 = 50 ✓

What is a Formula?

Formula

A formula is a rule written with letters showing how one quantity depends on others. Substituting known values into a formula lets you calculate the unknown.

Examples You Already Know

Area of rectangle: A = l × wArea of trapezium: A = ½(a + b)hPythagoras: c² = a² + b²

Read the question carefully. The question tells you which letter stands for what. Don't assume — the same letter can mean different things in different formulae.

The 2026 AQA Formulae Sheet

Every AQA GCSE Maths paper in 2026 comes with a formulae sheet as an insert — in Paper 1 and both calculator papers. Knowing exactly what's on it tells you what to memorise and what to save brain-space for.

On the 2026 Higher Formulae Sheet (Given in the Exam)

Area of trapezium: A = ½(a + b)h

Volume of prism: V = area of cross-section × length

Circumference of circle: C = 2πr = πd

Area of circle: A = πr²

Quadratic formula [H]

Pythagoras: a² + b² = c²

Trig ratios: sin A = a/c, cos A = b/c, tan A = a/b

Sine rule, cosine rule, Area = ½ab sin C [H]

Compound interest: P(1 + r/100)ⁿ

Probability: P(A or B), and P(A and B) [H]

NOT on the Sheet — Memorise (or Given Inside a Question)

Volume / surface area of cone, sphere, pyramid — given in the specific question when used

Kinematics: v = u + at; s = ut + ½at²; v² = u² + 2as — given in the specific question when used

Gradient = (y₂ − y₁)/(x₂ − x₁); y = mx + c

Exact trig values at 0°, 30°, 45°, 60°, 90°

Speed = distance ÷ time; Density = mass ÷ volume

Mean from a frequency table

Interior angle sum of a polygon: (n − 2) × 180°

Exam Tip

Having a formula given doesn't help if you don't know which one to pick. Examiners repeatedly flag students choosing the wrong formula — sine rule when Pythagoras was needed, cosine rule when basic trig would do.

Worked Example: Unfamiliar Formula

Spec A2 says: "unfamiliar formulae will be given in the question." Kinematics is the classic example — the formula is printed inside the question, and your job is to substitute carefully.

02MCE Framework in Action [2 marks]

The formula v = u + at gives the final velocity v of an object with initial velocity u, acceleration a, and time t.

Find v when u = 3, a = −2 and t = 5.

M — Method. Substitute values into the given formula. Put the negative a in brackets. Apply BIDMAS — multiplication before addition.

C — Calculate. Show every line:

v = 3 + (−2)(5)v = 3 + (−10)v = 3 − 10v = −7

E — Evaluate. State the answer: v = −7. The negative value means the object has reversed direction — physically sensible when acceleration acts against the initial velocity.

Substitution into Functions: f(x)

Function notation is a common AQA substitution pattern. When the question writes f(x) = …, then asking for f(−2) just means "substitute x = −2 into the expression on the right."

Function Notation

f(x) means "the output of function f when the input is x." It is not f multiplied by x. To find f(a), substitute x = a into the expression for f(x).

Example: Given f(x) = 3x² − 5x + 2, find f(−2). [2 marks]

M. Replace every x with (−2). Keep the brackets.

f(−2) = 3(−2)² − 5(−2) + 2

C. Indices first, then multiplications.

= 3(4) − 5(−2) + 2= 12 − (−10) + 2= 12 + 10 + 2E.f(−2) = 24.

Examiner Alert

Two brackets matter here. (−2)² = 4 (from Rule 2), AND −5 × (−2) = +10 (negative × negative = positive). Both are easy marks to lose if brackets are skipped.

Scientific Formula: Density

Spec A2 explicitly covers scientific formulae. Density, speed, and pressure appear regularly on AQA Higher papers — and they test unit discipline as much as substitution.

The Density Formula

ρ = m ÷ Vwhere ρ (rho) is density, m is mass, V is volume. Typical units: g/cm³ or kg/m³.

Example: A metal block has mass 540 g and volume 200 cm³. Find its density. [2 marks]

M. Use ρ = m / V. Both quantities are already in matching units (g and cm³), so no conversion is needed. The answer will be in g/cm³.

ρ = 540 ÷ 200ρ = 2.7

E.Density = 2.7 g/cm³. Without the unit, this scores M1 but not the A1.

Check Your Units

Formulae only work if the units match. Substituting centimetres into a formula that expects metres gives an answer wrong by a factor of 100 — or 10,000 for area, or 1,000,000 for volume.

Unit Conversion Example

A trapezium has parallel sides a = 40 cm and b = 1.2 m, with perpendicular height h = 0.5 m. Find its area.

Before substituting — convert so every length is in the same unit.Convert: 40 cm = 0.4 m. Now a = 0.4 m, b = 1.2 m, h = 0.5 m.A = ½(0.4 + 1.2) × 0.5 = ½ × 1.6 × 0.5 = 0.4 m²Skipping the conversion step is one of the most common Paper 2 errors on geometry questions.

Examiner Alert

Dropped units: AQA mark schemes require the correct unit (cm, cm², cm³, m/s, g/cm³, etc.) on numerical answers in context. A correct number without its unit can score M1 but not the A1.

Weak vs. Strong Student

Same maths. Same value of x. Different marks. Here's exactly what separates a full-mark answer from a partial one.

❌ Weak Student

3 × −2² = 3 × −4 = −12

Marks: 0 / 2

No brackets on the negative → −2² read as −(2²) = −4. Wrong sign. Final answer wrong.

✅ Strong Student

3 × (−2)² = 3 × 4 = 12

Marks: 2 / 2

Brackets around the −2 → (−2)² = 4. Full working shown. Correct answer.

The Golden Rule

Brackets cost nothing. Missing them costs marks.

Common Errors to Avoid

Students Often WriteCorrect VersionWhy It Matters

x = −3 into x² gives −9(−3)² = 9 (negative squared = positive)Sign error → wrong answer

3 × −2² = −123 × (−2)² = 12Missing brackets on negative

f(x) read as f × xf(x) = the function f evaluated at xChanges the whole calculation

Skip BIDMAS after substitutingIndices before multiplication; multiplication before additionOrder of operations applies throughout

Drop the unit on the final answerState the unit (cm, cm³, m/s, g/cm³, etc.)A1 mark lost — answer without unit is incomplete

Mix cm and m in one formulaConvert everything to the same unit before substitutingFactor-of-100 error on final answer

Write numbers straight down without showing substitutionShow the formula → substitute on the next line → evaluateMethod marks (M1) awarded for seen substitution

The MCE Framework

Every multi-mark substitution question is scored using the MCE framework. Method, Calculate, Evaluate — this is how examiners award the marks on a typical question.

MMethod. State the formula and which values you are substituting.

CCalculate. Show each line of working. Brackets on negatives. BIDMAS throughout.

EEvaluate. State the final numerical answer with the correct unit.

Typical Mark Allocation (from AQA Mark Schemes)

M1 — correct substitution shown (e.g. "3 × (−2)² seen")

A1 — final numerical answer (with correct unit where relevant)

2 marks total is the standard pattern. More marks appear when rearrangement or multi-step calculation is involved.

Examiner Alerts — The Big Ones

Missing Brackets on Negative Substitutions

What students write: they evaluate 3(−2)² as 3 × −4 = −12.

Why it loses marks: without brackets around the squared term, the minus sign gets stripped off. The working shows the student doesn't understand that squaring a negative gives a positive.

What to write instead: always wrap negatives in brackets. (−2)² = 4 because (−2) × (−2) = 4. So 3(−2)² = 3 × 4 = 12.

Dropped Units

What students write: they substitute into a volume or density formula and write a correct number with no unit.

Why it loses marks: the context requires a unit. A density of "2.7" is meaningless; "2.7 g/cm³" earns the A1.

What to write instead: always check the question. If it's a length, area, volume, speed, or density, the A1 needs the correct unit alongside the number.

Exam Technique

Top Exam Tip

On every substitution question, write the formula first, then write the substituted version on the next line. This gives you the M1 for method even if you slip up in the arithmetic. Students who jump straight to calculation often lose the method mark when they make a small error.

Speed Strategy for Paper 1 (Non-Calculator)

Substitute negatives in brackets — always.

Evaluate powers first — (−2)² before anything else.

Work line by line — one inverse operation per line.

Final check: does the answer make sense? A negative length, negative area, or negative probability is a red flag.

Speed Strategy for Papers 2 & 3 (Calculator)

Type the whole substituted expression into the calculator with brackets around every negative.

Don't round midway — keep exact values until the final line.

Re-enter and double-check the final calculation — it's the most common place to mistype.

Practice Question

A function is defined by:

f(x) = 2x² − 4x + 5

Find the value of f(−3). [2 marks]

M

Substitute x = −3 into f(x) = 2x² − 4x + 5. Put the negative in brackets. Apply BIDMAS — square first, then multiplications, then addition.

C

f(−3) = 2(−3)² − 4(−3) + 5

C

(−3)² = 9, so f(−3) = 2(9) − 4(−3) + 5

C

= 18 − (−12) + 5

C

= 18 + 12 + 5

E

f(−3) = 35. Both brackets mattered: (−3)² = +9, and −4 × (−3) = +12. Missing either sign flip changes the answer.

Quick-Fire Checklist

Before you move on, you should be able to do all of these without pausing:

Can You...

Evaluate (−4)² without writing −16? (Answer: 16)

Substitute a = 2, b = −3 into 2a² − b? (Answer: 11)

Find f(−1) when f(x) = x² + 4x − 2? (Answer: −5)

Use ρ = m/V with m = 120 g, V = 50 cm³? (Answer: 2.4 g/cm³)

Spot when a formula needs unit conversion before substituting?

Write the formula, the substitution, and the answer as three separate lines in your working?

One-Line Takeaway

Substitution = replace letters + brackets on negatives + BIDMAS + state the answer with units. Get those four things right every time and substitution questions become free marks.

Summary

Key Points

Substitution — replace every letter with its value, every time.

Always put negative values in brackets before evaluating.

BIDMAS — indices first, then multiplication, then addition/subtraction.

2026 formulae sheet — gives you trapezium area, prism volume, circle formulae, Pythagoras, trig, quadratic formula, compound interest, probability. Everything else you memorise — or it's given in the question.

Function notation — f(a) means substitute x = a into f(x). Not f × a.

Units — convert before substituting; state the unit in the final answer.

MCE — show method, calculation line-by-line, and a clearly stated evaluated answer.

MCE Framework Recap

M — Method: state the formula and what you are substituting.

C — Calculate: show every line; brackets on negatives; no skipped steps.

E — Evaluate: state the final answer with units; sense-check it.

Exam Tip

The most common way to lose marks on substitution questions is missing brackets on negative values. Whenever you substitute, write the negative in brackets — every single time — and you'll never drop the sign.