[discia] GCSE Maths › Algebra › Solving Quadratics

Solving Quadratics

discia.org 10 questions · 31 marks
Name Date
  1. Either paper 2 marks

    Here is the graph of \(y = 0.5x^2 + 3x - 2\)

    xy-8-7-6-5-4-3-212-8-7-6-5-4-3-2-112340

    Use the graph to estimate the solutions of \(0.5x^2 + 3x - 2 = 0\)

    Show answer
    \(x \approx -6.6\) and \(x \approx 0.6\)
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  2. Either paper 2 marks

    Solve \(x^2 + 6x + 5 = 0\)

    Show answer
    \(x = -5\) or \(x = -1\)
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  3. Either paper 2 marks

    \(x = 3\) is a solution of the equation

    \(x^2 +x + c = 0\)

    Work out the value of \(c\).

    Show answer
    \(-12\)
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  4. Either paper Higher4 marks

    Meera is using a graphical method to solve \(x^2 - 4x = 0\)

    She draws the graph of \(y = x^2\) and a straight line on the same grid.

    Here is the graph of \(y = x^2\)

    xy-6-5-4-3-2123456-2246810121416180
    (a)

    Complete her method to solve \(x^2 - 4x = 0\)

    Write down the equation of the straight line she draws and use the graph to find the solutions.

    (b)

    Levi is solving \(5x^2 - 8x = 0\)

    He uses this method.

    \(5x^2 - 8x = 0\)

    Add \(8x\) to both sides:

    \(5x^2 = 8x\)

    Divide both sides by \(x\):

    \(5x = 8\)

    Divide both sides by \(5\):

    \(x = 1.6\)

    Evaluate his method and his answer.

    Show answer
    (a) the line is \(y = 4x\); the solutions are \(x = 0\) and \(x = 4\)
    (b) Levi is wrong to divide by \(x\), because \(x\) can be \(0\), so he loses the solution \(x = 0\); his \(x = 1.6\) is a solution but the full answer is \(x = 0\) and \(x = 1.6\)
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  5. Either paper Higher3 marks

    Solve \(x^2 = 9x - 14\)

    Show answer
    \(x = 2\) or \(x = 7\)
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  6. Either paper Higher4 marks

    The width of a rectangle is \(x\) cm.

    The length of the rectangle is 2 cm more than its width.

    The area of the rectangle is 63 cm².

    Find the width of the rectangle.

    Show answer
    \(7\ \text{cm}\)
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  7. Either paper 3 marks

    The solutions of the equation

    \(x^2 + bx + c = 0\)

    are \(x = 2\) and \(x = 8\).

    Calculate the value of \(b\) and the value of \(c\).

    Show answer
    \(b = -10\) and \(c = 16\)
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  8. Either paper 5 marks

    The diagram shows the plan of the floor of an L-shaped room. All lengths are in metres.

    (x + 4) mx m(x + 4) m4 mDiagram NOT accurately drawn

    The area of the floor is 169 m².

    (a)

    Show that \(x^2 + 8x - 153 = 0\)

    (b)

    By factorising \(x^2 + 8x - 153\), work out the value of \(x\). You must show your working.

    Show answer
    (a) \(x(x + 4) + 4(x + 4) = 169\), so \(x^2 + 8x - 153 = 0\)
    (b) \(9\ \text{m}\)
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  9. Either paper Higher4 marks

    Jo wants to work out the solutions of \(x^2 + 9x + 2 = 0\)

    She says, "This equation has no solutions."

    (a)

    Is Jo correct? Tick a box.

    \(\square\) Yes \(\square\) No

    Give a reason for your answer.

    (b)

    Without expanding any brackets, show how to work out the exact solutions of

    \(9(x + 2)^2 = 25\)

    Give the solutions.

    Show answer
    (a) No. Here \(b^2 - 4ac = 73\), which is positive, so the equation has two solutions, found by the quadratic formula or by completing the square.
    (b) \(x = -\frac{1}{3}\) or \(x = -\frac{11}{3}\)
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  10. Either paper 2 marks

    Isla is asked to solve \(4t^2 + 5t - 6 = 0\)

    Here is her working and answer.

    \((4t - 3)(t + 2) = 0\)

    \(t = \frac{3}{4}\) or \(t = 2\)

    (a)

    What mistake has Isla made?

    (b)

    Work out the correct solutions of the equation.

    Show answer
    (a) Isla got the sign of the root wrong in the bracket with just t. The bracket that Isla solved to get \(t = 2\) gives \(t = -2\).
    (b) \(t = -2\) or \(t = \frac{3}{4}\)
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discia.org Solving Quadratics · sheet 1
Answers Solving Quadratics · sheet 1
  1. \(x \approx -6.6\) and \(x \approx 0.6\)
  2. \(x = -5\) or \(x = -1\)
  3. \(-12\)
  4. (a) the line is \(y = 4x\); the solutions are \(x = 0\) and \(x = 4\)
    (b) Levi is wrong to divide by \(x\), because \(x\) can be \(0\), so he loses the solution \(x = 0\); his \(x = 1.6\) is a solution but the full answer is \(x = 0\) and \(x = 1.6\)
  5. \(x = 2\) or \(x = 7\)
  6. \(7\ \text{cm}\)
  7. \(b = -10\) and \(c = 16\)
  8. (a) \(x(x + 4) + 4(x + 4) = 169\), so \(x^2 + 8x - 153 = 0\)
    (b) \(9\ \text{m}\)
  9. (a) No. Here \(b^2 - 4ac = 73\), which is positive, so the equation has two solutions, found by the quadratic formula or by completing the square.
    (b) \(x = -\frac{1}{3}\) or \(x = -\frac{11}{3}\)
  10. (a) Isla got the sign of the root wrong in the bracket with just t. The bracket that Isla solved to get \(t = 2\) gives \(t = -2\).
    (b) \(t = -2\) or \(t = \frac{3}{4}\)
discia.org Solving Quadratics · sheet 1