Solve quadratic inequality

Solve an inequality of the form ax² + bx + c > 0 or ≤ 0 by factorising or finding roots and testing regions.

58 questions · Moderate -0.9

1.02g Inequalities: linear and quadratic in single variable
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CAIE P1 2006 November Q10
10 marks Moderate -0.8
10 The function f is defined by \(\mathrm { f } : x \mapsto x ^ { 2 } - 3 x\) for \(x \in \mathbb { R }\).
  1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 4\).
  2. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } - b\), stating the values of \(a\) and \(b\).
  3. Write down the range of f .
  4. State, with a reason, whether f has an inverse. The function g is defined by \(\mathrm { g } : x \mapsto x - 3 \sqrt { } x\) for \(x \geqslant 0\).
  5. Solve the equation \(\mathrm { g } ( x ) = 10\).
CAIE P1 2013 November Q10
10 marks Moderate -0.3
10 A curve has equation \(y = 2 x ^ { 2 } - 3 x\).
  1. Find the set of values of \(x\) for which \(y > 9\).
  2. Express \(2 x ^ { 2 } - 3 x\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and state the coordinates of the vertex of the curve. The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 2 x ^ { 2 } - 3 x \quad \text { and } \quad \mathrm { g } ( x ) = 3 x + k$$ where \(k\) is a constant.
  3. Find the value of \(k\) for which the equation \(\mathrm { gf } ( x ) = 0\) has equal roots.
CAIE P1 2013 November Q1
3 marks Easy -1.2
1 Solve the inequality \(x ^ { 2 } - x - 2 > 0\).
CAIE P1 2016 November Q3
6 marks Moderate -0.8
3 A curve has equation \(y = 2 x ^ { 2 } - 6 x + 5\).
  1. Find the set of values of \(x\) for which \(y > 13\).
  2. Find the value of the constant \(k\) for which the line \(y = 2 x + k\) is a tangent to the curve.
Edexcel P1 2023 June Q1
4 marks Moderate -0.8
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
Solve the inequality $$4 x ^ { 2 } - 3 x + 7 \geq 4 x + 9$$
Edexcel C1 2012 January Q3
6 marks Moderate -0.8
3. Find the set of values of \(x\) for which
  1. \(4 x - 5 > 15 - x\)
  2. \(x ( x - 4 ) > 12\)
Edexcel C1 2013 June Q5
6 marks Easy -1.2
5. Find the set of values of \(x\) for which
  1. \(2 ( 3 x + 4 ) > 1 - x\)
  2. \(3 x ^ { 2 } + 8 x - 3 < 0\)
Edexcel C1 Q1
3 marks Easy -1.2
  1. Solve the inequality \(10 + x ^ { 2 } > x ( x - 2 )\).
    (3)
Edexcel C1 2006 June Q2
4 marks Moderate -0.8
Find the set of values of \(x\) for which $$x ^ { 2 } - 7 x - 18 > 0 .$$
OCR C1 2007 January Q3
5 marks Easy -1.2
3 Solve the inequalities
  1. \(3 ( x - 5 ) \leqslant 24\),
  2. \(5 x ^ { 2 } - 2 > 78\).
OCR C1 2005 June Q1
4 marks Moderate -0.8
1 Solve the inequality \(x ^ { 2 } - 6 x - 40 \geqslant 0\).
OCR C1 2008 June Q7
7 marks Moderate -0.8
7 Solve the inequalities
  1. \(8 < 3 x - 2 < 11\),
  2. \(y ^ { 2 } + 2 y \geqslant 0\).
OCR MEI C1 Q2
3 marks Easy -1.2
2 Find the range of values of \(x\) for which \(x ^ { 2 } - 5 x + 6 \leq 0\).
OCR MEI C1 Q10
12 marks Moderate -0.8
10
  1. A quadratic function is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 8\).
    Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the points where it crosses the axes. Mark the lowest point on the curve, and give its coordinates.
  2. Solve the inequality \(x ^ { 2 } - 6 x + 8 < 0\).
  3. On the same graph, sketch \(y = \mathrm { f } ( x + 3 )\).
  4. The graph of \(y = \mathrm { f } ( x + 3 ) - 2\) is obtained from the graph of \(y = \mathrm { f } ( x )\) by a transformation. Describe the transformation and sketch the curve on the same axes as in (i) and (iii) above. Label all these curves clearly.
OCR MEI C1 Q3
4 marks Moderate -0.5
3 Solve the inequality \(2 x ^ { 2 } - 7 x \geq 4\).
OCR C1 Q4
4 marks Moderate -0.5
4. Solve the inequality $$2 x ^ { 2 } - 9 x + 4 < 0 .$$
OCR C1 Q5
7 marks Moderate -0.8
  1. (i) Sketch on the same diagram the curve with equation \(y = ( x - 2 ) ^ { 2 }\) and the straight line with equation \(y = 2 x - 1\).
Label on your sketch the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the set of values of \(x\) for which $$( x - 2 ) ^ { 2 } > 2 x - 1$$
OCR C1 Q1
4 marks Moderate -0.3
  1. Solve the inequality
$$x ( 2 x + 1 ) \leq 6 .$$
OCR MEI C1 Q2
3 marks Moderate -0.8
2 Solve the inequality \(3 x ^ { 2 } + 10 x + 3 > 0\).
OCR MEI C1 Q3
4 marks Moderate -0.8
3 Solve the inequality \(5 x ^ { 2 } - 28 x - 12 \leqslant 0\).
OCR MEI C1 Q7
5 marks Easy -1.2
7 Solve the following inequalities.
  1. \(2 ( 1 - x ) > 6 x + 5\)
  2. \(( 2 x - 1 ) ( x + 4 ) < 0\)
OCR MEI C1 Q9
2 marks Easy -1.2
9 Solve the inequality \(x ( x - 6 ) > 0\).
OCR MEI C1 Q13
4 marks Easy -1.2
13 Solve the inequality \(x ^ { 2 } + 2 x < 3\).
OCR C1 2009 January Q8
10 marks Moderate -0.3
8
  1. Solve the equation \(5 - 8 x - x ^ { 2 } = 0\), giving your answers in simplified surd form.
  2. Solve the inequality \(5 - 8 x - x ^ { 2 } \leqslant 0\).
  3. Sketch the curve \(y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )\), giving the coordinates of the points where the curve crosses the coordinate axes.
OCR MEI C1 2013 January Q4
4 marks Moderate -0.8
4 Solve the inequality \(5 x ^ { 2 } - 28 x - 12 \leqslant 0\).