Mastery 2: Quadratic Expressions & Equations

Year 10 Mathematics

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Page 3 – Questions

Q1: Expand x(3 + x²)

Answer:

x(3 + x²) = 3x + x³

Marking:

• Correct expansion: 1 mark
• Final simplified form: 3x + x³

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Q2: Expand and simplify (x - 2)(2x + 5)(x + 3)

Working:

First, multiply (x - 2)(2x + 5):

(x)(2x) + (x)(5) + (-2)(2x) + (-2)(5) = 2x² + 5x - 4x - 10 = 2x² + x - 10

Then multiply (2x² + x - 10)(x + 3):

= 2x³ + 6x² + x² + 3x - 10x - 30

= 2x³ + 7x² - 7x - 30

Answer:

2x³ + 7x² - 7x - 30

Marking:

• Correct first product: 1 mark
• Correct final expansion and simplification: 1 mark
• Total: 2 marks

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Q3: Expand and simplify (x + 3)(x + 5)

Answer:

x² + 5x + 3x + 15 = x² + 8x + 15

Marking:

• Correct expansion and simplification: 1 mark

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Q4: Factorise completely x² + 8x + 15

Answer:

x² + 8x + 15 = (x + 3)(x + 5)

Marking:

• Correct factorisation: 1 mark

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Page 4 – Questions

Q5(a): Factorise completely 2m + 3p - 8km - 12kp

Working:

Group terms:

(2m - 8km) + (3p - 12kp) = 2m(1 - 4k) + 3p(1 - 4k)

= (1 - 4k)(2m + 3p)

Answer:

(1 - 4k)(2m + 3p)

Marking:

• Correct grouping and common factor extraction: 1 mark
• Final factorised form: 1 mark
• Total: 2 marks

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Q5(b): Factorise completely 5x² - 20y²

Working:

Factor out common factor 5:

5(x² - 4y²)

Then factorise difference of squares:

5(x - 2y)(x + 2y)

Answer:

5(x - 2y)(x + 2y)

Marking:

• Factor 5: 1 mark
• Factorise difference of squares: 1 mark
• Total: 2 marks

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Q6: Solve (5x - 3)(2x + 7) = 0

Answer:

Using zero-product property:

5x - 3 = 0 ⇒ x = ³⁄₅

2x + 7 = 0 ⇒ x = -⁷⁄₂

Marking:

• Each correct solution: 1 mark
• Total: 2 marks

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Q7: Factorise 6x² + 7x - 20

Working:

Find two numbers whose product is 6 × (-20) = -120 and sum is 7:

Numbers: 15 and -8

Rewrite middle term:

6x² + 15x - 8x - 20

Group:

3x(2x + 5) - 4(2x + 5)

= (2x + 5)(3x - 4)

Answer:

(2x + 5)(3x - 4)

Marking:

• Correct splitting of middle term: 1 mark
• Correct factorisation by grouping: 1 mark
• Total: 2 marks

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Page 5 – Questions

Q8(a): Factorise 18y - 3ay + 12x - 2ax

Working:

Group terms:

(18y - 3ay) + (12x - 2ax) = 3y(6 - a) + 2x(6 - a)

= (6 - a)(3y + 2x)

Answer:

(6 - a)(3y + 2x)

Marking:

• Correct grouping and factorisation: 1 mark

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Q8(b): Factorise x² - 36

Answer:

Difference of squares:

(x - 6)(x + 6)

Marking:

• Correct factorisation: 1 mark

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Q8(c): Factorise q² - 9

Answer:

Difference of squares:

(q - 3)(q + 3)

Marking:

• Correct factorisation: 1 mark