Chapter 4-21: Algebra Exam Technique

📝 Algebra Exam Technique (Higher Tier - Grades 6-9)

Algebra is worth 25-30% of your exam - the largest single topic area
Show all working clearly - algebra questions award method marks generously
Factorizing quadratics appears in every paper (3-4 marks each time)
Quadratic formula - memorize it! Not on formula sheet for some boards
Simultaneous equations - Grade 6-7 staple, often 4-5 marks
Algebraic fractions - Higher only, frequently in Paper 3 (Grade 7-9)
Changing the subject - when letter appears twice, it's Grade 8-9
Proof questions - explain every step, use algebra notation

⏱️ Timed Practice Mode

45:00

Real Past Paper Questions (AQA/Edexcel/OCR)

Question 1 (AQA 2023 Paper 1) 3 marks 4-5 mins

Factorize fully:

x² - 9

✓ Mark Scheme

Recognition: This is difference of two squares (1 mark)

Formula: a² - b² = (a + b)(a - b)

Step 1: x² - 9 = x² - 3² (identifying 9 = 3²)

Step 2: (x + 3)(x - 3) (2 marks for correct factorization)

Answer: (x + 3)(x - 3)

⚠️ Common Mistakes (Examiner Reports)

Writing (x + 3)² or (x - 3)² instead of (x + 3)(x - 3)
Not recognizing it as difference of two squares
Trying to factorize as (x + a)(x + b) when there's no x term

Question 2 (Edexcel 2023 Paper 2) 4 marks 6-8 mins

Solve:

x² + 5x - 14 = 0

✓ Mark Scheme

Method 1 - Factorizing (preferred for this question):

Step 1: Find two numbers that multiply to -14 and add to +5 (1 mark)

Numbers are +7 and -2 (since 7 × -2 = -14 and 7 + (-2) = 5)

Step 2: (x + 7)(x - 2) = 0 (1 mark)

Step 3: x + 7 = 0 or x - 2 = 0 (1 mark)

Step 4: x = -7 or x = 2 (1 mark)

Answer: x = -7 or x = 2

⚠️ Common Mistakes (Examiner Reports)

Getting signs wrong: (x - 7)(x + 2) gives x² - 5x - 14 (not the same!)
Only giving one solution (must give both x values)
Writing x = 7 and x = -2 (forgetting to change signs)
Not checking by expanding brackets

Question 3 (OCR 2023 Paper 3) 5 marks 8-10 mins

Solve using the quadratic formula:

2x² + 7x - 4 = 0

Give your answers to 2 decimal places.

x = (-b ± √(b² - 4ac)) / 2a

✓ Mark Scheme

Step 1: Identify a = 2, b = 7, c = -4 (1 mark)

Step 2: Calculate discriminant: b² - 4ac = 7² - 4(2)(-4) = 49 + 32 = 81 (1 mark)

Step 3: x = (-7 ± √81) / (2 × 2) = (-7 ± 9) / 4 (1 mark)

Step 4: x = (-7 + 9) / 4 = 2/4 = 0.5 (1 mark)

Step 5: x = (-7 - 9) / 4 = -16/4 = -4 (1 mark)

Answer: x = 0.50 or x = -4.00

⚠️ Common Mistakes (Examiner Reports)

Forgetting to double the 'a' in denominator: using 2a not just a
Sign errors: -4ac when c is negative becomes +32 not -32
Not giving both solutions (+ and - from ±)
Incorrect rounding to 2 d.p. (must show both as 0.50 and -4.00)
Writing answers as fractions when question asks for decimals

Question 4 (AQA 2022 Paper 3) 5 marks 8-10 mins

Solve the simultaneous equations:

3x + 2y = 16
5x - 3y = 3

✓ Mark Scheme (Elimination Method)

Step 1: Multiply first equation by 3 and second by 2 to eliminate y (1 mark)

9x + 6y = 48 ... (equation A)

10x - 6y = 6 ... (equation B)

Step 2: Add equations: 19x = 54 (1 mark)

Step 3: x = 54/19 (accept as fraction or 2.84...) (1 mark)

Step 4: Substitute into first equation: 3(54/19) + 2y = 16 (1 mark)

162/19 + 2y = 16

2y = 16 - 162/19 = 304/19 - 162/19 = 142/19

y = 71/19 (or 3.74...)

Answer: x = 54/19, y = 71/19 (or x = 2.84, y = 3.74) (1 mark for correct y)

⚠️ Common Mistakes (Examiner Reports)

Multiplying only one equation (need to match coefficients)
Sign errors when adding/subtracting equations
Not substituting back to find second variable
Arithmetic errors with fractions
Not checking answer works in both original equations

Question 5 (Edexcel 2023 Paper 3 - Grade 7-9) 4 marks 6-8 mins

Simplify fully:

(x² - 4) / (x² + 5x + 6)

✓ Mark Scheme

Step 1: Factorize numerator: x² - 4 = (x + 2)(x - 2) (1 mark - difference of squares)

Step 2: Factorize denominator: x² + 5x + 6 = (x + 2)(x + 3) (1 mark)

Find two numbers that multiply to 6 and add to 5: that's 2 and 3

Step 3: Write as: [(x + 2)(x - 2)] / [(x + 2)(x + 3)] (1 mark)

Step 4: Cancel common factor (x + 2): (x - 2) / (x + 3) (1 mark)

Answer: (x - 2) / (x + 3)

⚠️ Common Mistakes (Examiner Reports)

Not factorizing fully before attempting to cancel
Canceling terms instead of factors: x² cancels with x² (WRONG!)
Incorrect factorization of denominator
Leaving answer as factored form without canceling
Attempting to cancel numbers across numerator/denominator incorrectly

Question 6 (AQA 2023 Paper 3 - Grade 8-9) 6 marks 10-12 mins

Make t the subject of the formula:

v = u + at²

✓ Mark Scheme

Step 1: Subtract u from both sides (1 mark)

v - u = at²

Step 2: Divide both sides by a (1 mark)

(v - u) / a = t²

Step 3: Square root both sides (1 mark)

t = ±√[(v - u) / a]

Alternative acceptable form:

t = ±√(v - u) / √a or t = ±[√(v - u)] / √a (3 marks for any correct form)

Answer: t = ±√[(v - u) / a]

⚠️ Common Mistakes (Examiner Reports)

Forgetting ± when square rooting (must include both positive and negative)
Incorrectly splitting the square root: √[(v-u)/a] ≠ √(v-u)/a
Order of operations: must subtract u before dividing by a
Not keeping (v - u) in brackets when dividing by a

📊 What Different Grades Look Like (Higher Tier)

Grade 5-6:

• Can factorize simple quadratics where a = 1 (Question 2)

• Recognizes difference of two squares (Question 1)

• Can solve simultaneous equations by elimination (Question 4)

• Shows clear working and checks answers

Grade 7-8:

• Uses quadratic formula correctly with a ≠ 1 (Question 3)

• Simplifies algebraic fractions by factorizing (Question 5)

• Solves harder simultaneous equations with non-integer solutions

• Makes subject of formula with basic rearrangement

Grade 9:

• Makes subject when letter appears twice or under square root (Question 6)

• Solves simultaneous equations with one linear, one quadratic

• Completes algebraic proof questions with clear reasoning

• Works confidently with complex algebraic fractions

• Remembers to include ± when appropriate

🎯 Grade 8-9 Algebra Tips

Always check your answer - substitute back into original equation
Quadratic formula: Write it out first, then substitute values carefully
Algebraic fractions: Factorize everything first, then cancel common factors
Proof questions: Use "Let n = ..." and show algebraic steps, not examples
Rearranging formulae: Treat other letters as you would numbers
When letter appears twice: Get all terms with that letter on one side first

⚡ Quick Revision Summary

Difference of two squares: a² - b² = (a + b)(a - b)
Quadratic formula: x = [-b ± √(b² - 4ac)] / 2a
Factorizing x² + bx + c: Find two numbers that multiply to c, add to b
Simultaneous equations: Multiply to match coefficients, then eliminate
Algebraic fractions: Factorize top and bottom, then cancel factors (not terms)
Changing subject: Inverse operations in reverse BIDMAS order
Square rooting: Remember ± (positive and negative)
Completing the square: x² + bx = (x + b/2)² - (b/2)²
Expanding (a + b)²: a² + 2ab + b² (NOT a² + b²!)
Method marks: Show every step clearly - can earn 70% even with wrong answer