Quadratic inequality solving

A question is this type if and only if it requires solving a quadratic inequality (e.g. 2x^2 - x - 3 > 0) and expressing the solution as a set of values or intervals, possibly after sketching the curve.

6 questions · Moderate -0.7

1.02g Inequalities: linear and quadratic in single variable
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OCR C1 2008 January Q6
8 marks Moderate -0.3
6
  1. Solve the equation \(x ^ { 2 } + 8 x + 10 = 0\), giving your answers in simplified surd form.
  2. Sketch the curve \(y = x ^ { 2 } + 8 x + 10\), giving the coordinates of the point where the curve crosses the \(y\)-axis.
  3. Solve the inequality \(x ^ { 2 } + 8 x + 10 \geqslant 0\).
OCR C1 2007 June Q8
9 marks Moderate -0.8
8
  1. Express \(x ^ { 2 } + 8 x + 15\) in the form \(( x + a ) ^ { 2 } - b\).
  2. Hence state the coordinates of the vertex of the curve \(y = x ^ { 2 } + 8 x + 15\).
  3. Solve the inequality \(x ^ { 2 } + 8 x + 15 > 0\).
OCR C1 2015 June Q8
9 marks Moderate -0.3
8
  1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\), giving the coordinates of all points of intersection with the axes.
  2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - x - 3 > 0\).
  3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of the constant \(k\).
OCR MEI AS Paper 2 2022 June Q2
4 marks Easy -1.2
2
  1. Factorise \(3 x ^ { 2 } - 19 x - 14\).
  2. Solve the inequality \(3 x ^ { 2 } - 19 x - 14 < 0\).
AQA C1 2009 January Q2
4 marks Moderate -0.8
2
  1. Factorise \(2 x ^ { 2 } - 5 x + 3\).
  2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - 5 x + 3 < 0\).
OCR MEI AS Paper 1 2021 November Q8
12 marks Moderate -0.8
8 In this question you must show detailed reasoning.
  1. Use differentiation to find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 3 x - 2\).
  2. Use the second derivative to determine the nature of the stationary point.
  3. Show by shading on a sketch the region defined by the inequality \(y \geqslant 2 x ^ { 2 } - 3 x - 2\), indicating clearly whether the boundary is included or not.
  4. Solve the inequality \(2 x ^ { 2 } - 3 x - 2 > 0\) using set notation for your answer.