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AQA MathsAlgebra

Algebraic fractions

Simplify and calculate with fractions containing algebra.

AQAGCSE MathsAlgebraHigher
Visual model

Factorise first, then cancel common factors

(x+2)(x+3)(x+2)(x+3)(x+2)(x+2)cancel factors, not loose terms
Gold-standard guide
26 mins

What you will learn

Simplify and calculate with fractions containing algebra.
Use a clear step-by-step method for algebraic fractions.
Check your answer and avoid the most common exam mistake.
Useful before you start
Core number skillsEarlier algebra skillsShowing clear working
Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Method

To simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors

Step 1

Factorise the numerator using the difference of two squares

x2 − 9 = (x + 3)(x − 3)

Step 2

Factorise the denominator

x2 − x − 6 = (x − 3)(x + 2)

Step 3

Cancel the common factor (x − 3)

(x + 3)(x − 3) / (x − 3)(x + 2) = (x + 3)/(x + 2)

Watch out

Watch out

Students try to cancel across addition signs, writing (x2 − 9)/(x2 − x − 6) and cancelling x2

f
Simplify

Factorise first, then cancel common factors.

f
Solve

Multiply through by the denominators, checking excluded values.

Worked example

Simplify (x2 − 9)/(x2 − x − 6).

1

Factorise the numerator using the difference of two squares: x2 − 9 = (x + 3)(x − 3).

2

Factorise the denominator: x2 − x − 6 = (x − 3)(x + 2). Pair that multiplies to −6 and adds to −1 is (−3, +2).

3

Cancel the common factor (x − 3): (x + 3)(x − 3) / (x − 3)(x + 2) = (x + 3)/(x + 2).

4

State any excluded values: x ≠ 3 (denominator of original would be zero), x ≠ −2 (denominator of result would be zero).

Final answer

(x + 3)/(x + 2)

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Simplify (x2 − 9)/(x2 − x − 6).

4 marks4 minsalgebraic-fractions-worked
Show solution
Worked solution
  1. 1.Factorise the numerator using the difference of two squares: x2 − 9 = (x + 3)(x − 3).
  2. 2.Factorise the denominator: x2 − x − 6 = (x − 3)(x + 2). Pair that multiplies to −6 and adds to −1 is (−3, +2).
  3. 3.Cancel the common factor (x − 3): (x + 3)(x − 3) / (x − 3)(x + 2) = (x + 3)/(x + 2).
  4. 4.State any excluded values: x ≠ 3 (denominator of original would be zero), x ≠ −2 (denominator of result would be zero).
Final answer

(x + 3)/(x + 2)

Mark points
  • M1: factorise the numerator using the difference of two squares
  • M1: factorise the denominator
  • M1: cancel the common factor (x − 3)
  • M1: state any excluded values
  • A1: (x + 3)/(x + 2)
Watch out

Students try to cancel across addition signs, writing (x2 − 9)/(x2 − x − 6) and cancelling x2.Cancelling only works for factors in multiplication.Factorising first is the only safe way to identify what can be cancelled.

Diagnosticrecall

Simplify (x2 − 4)/(x + 2).

1 mark2 minsalgebraic-fractions-q1
Show solution
Worked solution
  1. 1.Spot the skill: To simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.
  2. 2.Use the factorise the numerator using the difference of two squares stage first, then factorise the denominator.
  3. 3.Keep the final answer visible: x − 2.
Final answer

x − 2

Mark points
  • M1: use the correct to simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.never cancel terms that are added or subtracted — only cancel factors (multiplied parts).for adding/subtracting: find a common denominator, convert each fraction, combine numerators.
  • A1: x − 2
Watch out

Students try to cancel across addition signs, writing (x2 − 9)/(x2 − x − 6) and cancelling x2.Cancelling only works for factors in multiplication.Factorising first is the only safe way to identify what can be cancelled.

Easyprocedure

Simplify (2x2 + 3x)/(4x2 − x).

2 marks3 minsalgebraic-fractions-q2
Show solution
Worked solution
  1. 1.Spot the skill: To simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.
  2. 2.Use the factorise the denominator stage first, then cancel the common factor (x − 3).
  3. 3.Keep the final answer visible: (2x + 3)/(4x − 1).
Final answer

(2x + 3)/(4x − 1)

Mark points
  • M1: use the correct to simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.never cancel terms that are added or subtracted — only cancel factors (multiplied parts).for adding/subtracting: find a common denominator, convert each fraction, combine numerators.
  • A1: (2x + 3)/(4x − 1)
Watch out

Students try to cancel across addition signs, writing (x2 − 9)/(x2 − x − 6) and cancelling x2.Cancelling only works for factors in multiplication.Factorising first is the only safe way to identify what can be cancelled.

Mediumreasoning

Add 1/(x + 1) + 2/(x − 1). Simplify fully.

3 marks4 minsalgebraic-fractions-q3
Show solution
Worked solution
  1. 1.Spot the skill: To simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.
  2. 2.Use the cancel the common factor (x − 3) stage first, then state any excluded values.
  3. 3.Keep the final answer visible: (3x − 1)/((x + 1)(x − 1)).
Final answer

(3x − 1)/((x + 1)(x − 1))

Mark points
  • M1: use the correct to simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.never cancel terms that are added or subtracted — only cancel factors (multiplied parts).for adding/subtracting: find a common denominator, convert each fraction, combine numerators.
  • A1: (3x − 1)/((x + 1)(x − 1))
Watch out

Students try to cancel across addition signs, writing (x2 − 9)/(x2 − x − 6) and cancelling x2.Cancelling only works for factors in multiplication.Factorising first is the only safe way to identify what can be cancelled.

Hardproblem solving

Solve 3/(x − 2) = 5/(x + 1).

3 marks5 minsalgebraic-fractions-q4
Show solution
Worked solution
  1. 1.Spot the skill: To simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.
  2. 2.Use the state any excluded values stage first, then factorise the numerator using the difference of two squares.
  3. 3.Keep the final answer visible: x = 1121\frac{1}{2}.
Final answer

x = 1121\frac{1}{2}

Mark points
  • M1: use the correct to simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.never cancel terms that are added or subtracted — only cancel factors (multiplied parts).for adding/subtracting: find a common denominator, convert each fraction, combine numerators.
  • A1: x = 1121\frac{1}{2}
Watch out

Students try to cancel across addition signs, writing (x2 − 9)/(x2 − x − 6) and cancelling x2.Cancelling only works for factors in multiplication.Factorising first is the only safe way to identify what can be cancelled.

Exam-stylemulti-step

Simplify (x2 + 5x + 6)/(x2 + 4x + 3).

4 marks6 minsalgebraic-fractions-q5
Show solution
Worked solution
  1. 1.Spot the skill: To simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.
  2. 2.Use the factorise the numerator using the difference of two squares stage first, then factorise the denominator.
  3. 3.Keep the final answer visible: (x + 2)/(x + 1).
Final answer

(x + 2)/(x + 1)

Mark points
  • M1: use the correct to simplify algebraic fractions: factorise numerator and denominator fully, then cancel common factors.never cancel terms that are added or subtracted — only cancel factors (multiplied parts).for adding/subtracting: find a common denominator, convert each fraction, combine numerators.
  • A1: (x + 2)/(x + 1)
Watch out

Students try to cancel across addition signs, writing (x2 − 9)/(x2 − x − 6) and cancelling x2.Cancelling only works for factors in multiplication.Factorising first is the only safe way to identify what can be cancelled.

Grade 9 stretchproblem solving

Solve 3/(x + 1) = 2/(x - 2).

4 marks7 minsalg-frac-g9
Show solution
Worked solution
  1. 1.Cross multiply.
  2. 2.Expand and solve the linear equation.
  3. 3.Check excluded values.
Final answer

x = 8

Mark points
  • M1: 3(x - 2) = 2(x + 1)
  • A1: x = 8
Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint
16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Algebraic fractions - 2 marksSimplify (x2 − 4)/(x + 2).Mark answer
Answer

x − 2

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer
Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer
Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer
Answer

3x(2x + 3)

Mastery check
  • I can explain the method for algebraic fractions.
  • I can show clear working without skipping key steps.
  • I can avoid this mistake: Students try to cancel across addition signs, writing (x2 − 9)/(x2 − x − 6) and cancelling x2.Cancelling only works for factors in multiplication.Factorising first is the only safe way to identify what can be cancelled.
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Related topics
Algebraic notation and substitutionSimplifying expressionsExpanding and factorising
Official exam-board sources

This guide follows the AQA GCSE Mathematics 8300 specification. Practice questions are original Learnova questions shaped around official content and exam skills.

Official specification Official past papers
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