Solve quadratic inequality

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

51 questions · Moderate -0.9

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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)
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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
3 marks Moderate -0.8
  1. (i) Calculate the discriminant of \(2 x ^ { 2 } + 8 x + 8\).
    (ii) State the number of real roots of the equation \(2 x ^ { 2 } + 8 x + 8 = 0\).
  2. Find the set of values of \(x\) for which
$$( x - 1 ) ( x - 2 ) < 20 .$$
OCR C1 Q1
4 marks Moderate -0.3
  1. Solve the inequality
$$x ( 2 x + 1 ) \leq 6 .$$
OCR C1 Q3
5 marks Moderate -0.3
  1. Find the set of values of \(x\) for which
    1. \(6 x - 11 > x + 4\),
    2. \(x ^ { 2 } - 6 x - 16 < 0\).
    3. (i) Sketch on the same diagram the graphs of \(y = ( x - 1 ) ^ { 2 } ( x - 5 )\) and \(y = 8 - 2 x\).
    Label on your diagram the coordinates of any points where each graph meets the coordinate axes.
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$( x - 1 ) ^ { 2 } ( x - 5 ) = 8 - 2 x$$
  3. State the integer, \(n\), such that $$n < \alpha < n + 1 .$$
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 \(x ^ { 2 } + 2 x < 3\).