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.

17 questions · Moderate -0.4

1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable
Sort by: Default | Easiest first | Hardest first
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 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 \(k\).