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Quadratic Inequalities

discia.org 10 questions · 25 marks
Name Date
  1. Either paper 2 marks

    \(x^2 - 11x + 28 \lt 0\)

    Is \(x = 6\) a solution of this inequality?

    You must show your working.

    Show answer
    Yes. When \(x = 6\) the expression is equal to \(-2\), which is less than zero.
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  2. Either paper Higher1 mark

    Here is a sketch of \(y = f(x)\) where \(f(x)\) is a quadratic function.

    The graph crosses the \(x\)-axis where \(x = -2.5\) and \(x = -1\)

    xy-2.5-1

    Circle the solution of \(f(x) > 0\)

    \(x < -2.5 \text{ or } x > -1\)

    \(-2.5 < x < -1\)

    \(x < -2.5 \text{ or } x < -1\)

    \(x > -2.5 \text{ or } x > -1\)

    Show answer
    \(-2.5 < x < -1\)
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  3. Either paper Higher1 mark

    Write down the solution of \(x^2 \gt 25\)

    Show answer
    \(x \lt -5\) or \(x \gt 5\)
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  4. Either paper Higher4 marks

    The graph of \(y = x^2 - 6x + 5\) is drawn on the grid.

    xy-3-2123456789-6-5-4-3-21234567891011120
    (a)

    Use the graph to solve \(x^2 - 6x + 5 < 0\)

    (b)

    Use the graph to solve \(x^2 - 6x + 5 > 0\)

    Show answer
    (a) \(1 < x < 5\)
    (b) \(x < 1\) or \(x > 5\)
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  5. Either paper Higher3 marks

    Solve \(4x^2 \gt 64\)

    Show answer
    \(x \lt -4\) or \(x \gt 4\)
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  6. Either paper 1 mark

    Yusuf has solved the inequality \(x^2 \gt 49\) and shown the solution on the number line.

    -9-8-7-6-5-4-3-2-10123456789

    Give a value of \(x\) that shows Yusuf's solution is wrong, and say why.

    Show answer
    For example \(x = 8\): it is not shaded on the line, but \(8^2 = 64\) is more than 49, so it should be shaded. Any value that is shaded when it does not work, or not shaded when it works, is correct. The values that work are outside \(-7\) and \(7\).
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  7. Either paper Higher3 marks

    Solve \(x^2 -x - 6 < 0\)

    Show answer
    \(-2 < x < 3\)
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  8. Either paper Higher3 marks

    Solve the inequality \(x^2 + 8x + 15 \geq 0\)

    Show answer
    \(x \leq -5\) or \(x \geq -3\)
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  9. Either paper Higher3 marks

    List all the integers that satisfy \(x^2 + 8x + 7 \leq 0\)

    Show answer
    \(-7,\ -6,\ -5,\ -4,\ -3,\ -2,\ -1\)
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  10. Either paper Higher4 marks

    Solve \(x^2 > -5x + 14\)

    Show answer
    \(x < -7\) or \(x > 2\)
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discia.org Quadratic Inequalities · sheet 1
Answers Quadratic Inequalities · sheet 1
  1. Yes. When \(x = 6\) the expression is equal to \(-2\), which is less than zero.
  2. \(-2.5 < x < -1\)
  3. \(x \lt -5\) or \(x \gt 5\)
  4. (a) \(1 < x < 5\)
    (b) \(x < 1\) or \(x > 5\)
  5. \(x \lt -4\) or \(x \gt 4\)
  6. For example \(x = 8\): it is not shaded on the line, but \(8^2 = 64\) is more than 49, so it should be shaded. Any value that is shaded when it does not work, or not shaded when it works, is correct. The values that work are outside \(-7\) and \(7\).
  7. \(-2 < x < 3\)
  8. \(x \leq -5\) or \(x \geq -3\)
  9. \(-7,\ -6,\ -5,\ -4,\ -3,\ -2,\ -1\)
  10. \(x < -7\) or \(x > 2\)
discia.org Quadratic Inequalities · sheet 1