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Factorising Harder Quadratics

discia.org 10 questions · 25 marks
Name Date
  1. Either paper Higher2 marks

    Factorise completely \(27x^2 - 192\)

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    \(3(3x + 8)(3x - 8)\)
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  2. Either paper 2 marks

    \(20v^2 - 21v - 54 = (4v - 9)(gv + h)\)

    This is true for all values of \(v\).

    (a)

    Work out the value of \(g\).

    (b)

    Work out the value of \(h\).

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    (a) \(5\)
    (b) \(6\)
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  3. Either paper Higher2 marks

    Factorise \(15w^2 + 28w + 12\)

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    \((3w + 2)(5w + 6)\)
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  4. Either paper Higher3 marks

    Write \(15y^2 - 48y - 48\) as a product of three factors.

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    \(3(5y + 4)(y - 4)\)
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  5. Either paper Higher3 marks

    Solve the equation \(12x^2 - 13x - 35 = 0\) by factorising.

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    \(x = -\frac{5}{4}\) or \(x = \frac{7}{3}\)
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  6. Either paper Higher4 marks

    The \(n\)th term of a sequence is \(4n^2 + 21n - 49\)

    (a)

    Factorise fully \(4n^2 + 21n - 49\)

    (b)

    Is any term of the sequence a prime number? Tick a box.

    Yes / No

    Give a reason for your answer.

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    (a) \((4n - 7)(n + 7)\)
    (b) No. The second bracket \(n + 7\) is always greater than 1, so a term is prime only if the first bracket \(4n - 7\) is 1. At \(n = 2\) the brackets are \(1\) and \(9\), and the first bracket changes by \(4\) each time \(n\) goes up by 1: the first bracket is 1 only for this one value of n and the second bracket is not prime, so no term is prime.
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  7. Either paper 3 marks
    (a)

    Factorise \(a^2 - b^2\)

    (b)

    Hence show that \(2^{ 214 } - 1\) is the product of two consecutive odd numbers.

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    (a) \((a - b)(a + b)\)
    (b) \(2^{ 214 } - 1 = (2^{ 107 } - 1)(2^{ 107 } + 1)\); \(2^{ 107 }\) is even, so \(2^{ 107 } - 1\) and \(2^{ 107 } + 1\) are both odd, and they differ by \(2\), so these are consecutive odd numbers.
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  8. Either paper Higher2 marks

    Factorise \(98t^2 - 8\) completely.

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    \(2(7t + 2)(7t - 2)\)
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  9. Either paper 2 marks

    \(25v^2 - 70v + 48 = (5v - 8)(gv + h)\)

    This is true for all values of \(v\).

    (a)

    Calculate the value of \(g\).

    (b)

    Calculate the value of \(h\).

    Show answer
    (a) \(5\)
    (b) \(-6\)
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  10. Either paper Higher2 marks

    Factorise fully \(18m^2 - 33m + 5\)

    Show answer
    \((3m - 5)(6m - 1)\)
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discia.org Factorising Harder Quadratics · sheet 1
Answers Factorising Harder Quadratics · sheet 1
  1. \(3(3x + 8)(3x - 8)\)
  2. (a) \(5\)
    (b) \(6\)
  3. \((3w + 2)(5w + 6)\)
  4. \(3(5y + 4)(y - 4)\)
  5. \(x = -\frac{5}{4}\) or \(x = \frac{7}{3}\)
  6. (a) \((4n - 7)(n + 7)\)
    (b) No. The second bracket \(n + 7\) is always greater than 1, so a term is prime only if the first bracket \(4n - 7\) is 1. At \(n = 2\) the brackets are \(1\) and \(9\), and the first bracket changes by \(4\) each time \(n\) goes up by 1: the first bracket is 1 only for this one value of n and the second bracket is not prime, so no term is prime.
  7. (a) \((a - b)(a + b)\)
    (b) \(2^{ 214 } - 1 = (2^{ 107 } - 1)(2^{ 107 } + 1)\); \(2^{ 107 }\) is even, so \(2^{ 107 } - 1\) and \(2^{ 107 } + 1\) are both odd, and they differ by \(2\), so these are consecutive odd numbers.
  8. \(2(7t + 2)(7t - 2)\)
  9. (a) \(5\)
    (b) \(-6\)
  10. \((3m - 5)(6m - 1)\)
discia.org Factorising Harder Quadratics · sheet 1