Practice Maths

Topic Review — Algebra: Expanding and Factorising

Mixed Practice — L08 – L11

This review covers all lessons in the Expanding and Factorising topic. Try each question before checking your answers.

Review Questions

  1. Expanding expressions. Fluency

    1. Expand: 4(3x − 7)
    2. Expand: −2x(x + 5)
    3. Expand and simplify: (x + 6)(x − 2)
    4. Expand and simplify: (3a + 4)(2a − 3)
    5. Expand using a special product rule: (y + 7)2
    6. Expand using a special product rule: (5m − 3)(5m + 3)

  2. Factorising expressions. Fluency

    1. Factorise: 10x + 15
    2. Factorise: 6a2b − 4ab2
    3. Factorise: x2 + 6x + 9
    4. Factorise: x2 − 5x − 14
    5. Factorise: 4p2 − 25
    6. Factorise fully: 3x2 − 12x + 12

  3. Mixed expanding and factorising. Understanding

    1. Expand (2x − 5)2, then factorise your result back to verify.
    2. Factorise 2x2 + 14x + 24 by taking out the HCF first, then factorising the trinomial.
    3. Factorise using grouping: 3xy + 6x − 4y − 8
    4. A student claims x2 − 16 = (x − 4)2. Are they correct? Expand (x − 4)2 to explain.
    5. Without expanding, decide which expression is larger when x = 10: (x + 3)(x − 3) or (x − 3)2. Explain.

  4. Solving problems with algebra. Problem Solving

    1. Solve x2 − 7x + 10 = 0 by factorising. Show all steps.
    2. The area of a rectangle is (6x2 + 7x − 3) cm2. The width is (2x + 3) cm. By factorising, find the length of the rectangle.
    3. Use the identity (a + b)(a − b) = a2 − b2 to evaluate 203 × 197 without a calculator. Show your method.
    4. A right-angled triangle has legs of length x cm and (x + 7) cm and a hypotenuse of (x + 8) cm. Use Pythagoras’ theorem to form a quadratic equation. Solve by factorising to find the dimensions of the triangle.

  5. Algebraic reasoning and proof. Reasoning

    1. (x + 5) m (x + 5) m (x − 1) (x − 1) Shaded = ? cut A large square has side (x + 5) m and a small square of side (x − 1) m is cut from one corner. Show that the remaining shaded area simplifies to 12(x + 2) m². Hint: use the difference of two squares identity.
    2. Show that (2n + 1)² − 1 is always divisible by 8 for any integer n. Factorise the result completely to justify your answer.
    3. Alex writes 6x² − 6 = 6(x² − 1). Bailey writes 6x² − 6 = 6(x + 1)(x − 1). Who has fully factorised the expression? Explain, and write the fully factorised form.
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