Quadratic equation real roots

Find parameter values for which a quadratic equation has two distinct real roots, equal roots, or no real roots, using discriminant analysis.

19 questions · Moderate -0.4

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CAIE P1 2023 June Q3
3 marks Moderate -0.8
3
  1. Express \(4 x ^ { 2 } - 24 x + p\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\).
  2. Hence or otherwise find the set of values of \(p\) for which the equation \(4 x ^ { 2 } - 24 x + p = 0\) has no real roots.
CAIE P1 2022 November Q3
5 marks Moderate -0.8
3
  1. Find the set of values of \(k\) for which the equation \(8 x ^ { 2 } + k x + 2 = 0\) has no real roots.
  2. Solve the equation \(8 \cos ^ { 2 } \theta - 10 \cos \theta + 2 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
Edexcel C1 2007 January Q5
4 marks Moderate -0.3
5. The equation \(2 x ^ { 2 } - 3 x - ( k + 1 ) = 0\), where \(k\) is a constant, has no real roots. Find the set of possible values of \(k\).
Edexcel C1 2008 January Q8
7 marks Moderate -0.3
8. The equation $$x ^ { 2 } + k x + 8 = k$$ has no real solutions for \(x\).
  1. Show that \(k\) satisfies \(k ^ { 2 } + 4 k - 32 < 0\).
  2. Hence find the set of possible values of \(k\).
Edexcel C1 2009 January Q7
7 marks Moderate -0.8
7. The equation \(k x ^ { 2 } + 4 x + ( 5 - k ) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 5 k + 4 > 0 .$$
  2. Hence find the set of possible values of \(k\).
Edexcel C1 2011 January Q8
7 marks Moderate -0.8
8. The equation \(x ^ { 2 } + ( k - 3 ) x + ( 3 - 2 k ) = 0\), where \(k\) is a constant, has two distinct real roots.
  1. Show that \(k\) satisfies $$k ^ { 2 } + 2 k - 3 > 0$$
  2. Find the set of possible values of \(k\).
Edexcel C1 2013 January Q9
7 marks Moderate -0.3
9. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
  2. Hence find the set of possible values of \(k\).
Edexcel C1 2014 January Q8
7 marks Moderate -0.3
  1. The equation \(2 x ^ { 2 } + 2 k x + ( k + 2 ) = 0\), where \(k\) is a constant, has two distinct real roots.
    1. Show that \(k\) satisfies
    $$k ^ { 2 } - 2 k - 4 > 0$$
  2. Find the set of possible values of \(k\).
Edexcel C1 2007 June Q7
6 marks Moderate -0.8
7. The equation \(x ^ { 2 } + k x + ( k + 3 ) = 0\), where \(k\) is a constant, has different real roots.
  1. Show that \(k ^ { 2 } - 4 k - 12 > 0\).
  2. Find the set of possible values of \(k\).
Edexcel C1 2008 June Q8
5 marks Moderate -0.3
Given that the equation \(2 q x ^ { 2 } + q x - 1 = 0\), where \(q\) is a constant, has no real roots,
  1. show that \(q ^ { 2 } + 8 q < 0\).
  2. Hence find the set of possible values of \(q\).
Edexcel C1 2015 June Q5
7 marks Moderate -0.3
  1. The equation
$$( p - 1 ) x ^ { 2 } + 4 x + ( p - 5 ) = 0 , \text { where } p \text { is a constant }$$ has no real roots.
  1. Show that \(p\) satisfies \(p ^ { 2 } - 6 p + 1 > 0\)
  2. Hence find the set of possible values of \(p\).
Edexcel C1 2018 June Q7
8 marks Moderate -0.3
  1. The equation \(20 x ^ { 2 } = 4 k x - 13 k x ^ { 2 } + 2\), where \(k\) is a constant, has no real roots.
    1. Show that \(k\) satisfies the inequality
    $$2 k ^ { 2 } + 13 k + 20 < 0$$
  2. Find the set of possible values for \(k\).
OCR C1 Q5
6 marks Moderate -0.8
5. Given that the equation $$x ^ { 2 } + 4 k x - k = 0$$ has no real roots,
  1. show that $$4 k ^ { 2 } + k < 0 ,$$
  2. find the set of possible values of \(k\).
OCR C1 2013 January Q8
7 marks Standard +0.3
8 The quadratic equation \(k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0\) has no real roots. Find the set of possible values of \(k\).
OCR C1 2016 June Q9
7 marks Standard +0.3
9 Find the set of values of \(k\) for which the equation \(x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )\) has two distinct real roots.
OCR MEI C1 2007 January Q8
4 marks Moderate -0.5
8 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 2 = 0\) has no real roots.
AQA C1 2007 January Q7
7 marks Moderate -0.3
7 The quadratic equation \(( k + 1 ) x ^ { 2 } + 12 x + ( k - 4 ) = 0\) has real roots.
  1. Show that \(k ^ { 2 } - 3 k - 40 \leqslant 0\).
  2. Hence find the possible values of \(k\).
AQA C1 2007 June Q7
7 marks Moderate -0.3
7 The quadratic equation $$( 2 k - 3 ) x ^ { 2 } + 2 x + ( k - 1 ) = 0$$ where \(k\) is a constant, has real roots.
  1. Show that \(2 k ^ { 2 } - 5 k + 2 \leqslant 0\).
    1. Factorise \(2 k ^ { 2 } - 5 k + 2\).
    2. Hence, or otherwise, solve the quadratic inequality $$2 k ^ { 2 } - 5 k + 2 \leqslant 0$$
AQA C1 2008 June Q8
7 marks Moderate -0.3
8 The quadratic equation \(( k + 1 ) x ^ { 2 } + 4 k x + 9 = 0\) has real roots.
  1. Show that \(4 k ^ { 2 } - 9 k - 9 \geqslant 0\).
  2. Hence find the possible values of \(k\).