Find range for no real roots

A question is this type if and only if it asks to find the set of values of a constant (k, p, m, etc.) for which a quadratic equation has no real roots, by setting the discriminant strictly less than zero.

24 questions · Moderate -0.4

1.02d Quadratic functions: graphs and discriminant conditions
Sort by: Default | Easiest first | Hardest first
Edexcel P1 2019 January Q9
7 marks Standard +0.3
  1. The equation
$$\frac { 3 } { x } + 5 = - 2 x + c$$ where \(c\) is a constant, has no real roots.
Find the range of possible values of \(c\).
Edexcel P1 2023 January Q4
4 marks Moderate -0.8
  1. Given that the equation \(k x ^ { 2 } + 6 k x + 5 = 0 \quad\) where \(k\) is a non zero constant has no real roots, find the range of possible values for \(k\).
Edexcel C12 2014 January Q8
7 marks Moderate -0.3
8. Find the range of values of \(k\) for which the quadratic equation $$k x ^ { 2 } + 8 x + 2 ( k + 7 ) = 0$$ has no real roots.
Edexcel C12 2019 January Q7
5 marks Moderate -0.3
7. The equation \(2 x ^ { 2 } + 5 p x + p = 0\), where \(p\) is a constant, has no real roots. Find the set of possible values for \(p\).
Edexcel C1 2006 January Q10
11 marks Moderate -0.3
10. $$x ^ { 2 } + 2 x + 3 \equiv ( x + a ) ^ { 2 } + b .$$
  1. Find the values of the constants \(a\) and \(b\).
  2. In the space provided below, sketch the graph of \(y = x ^ { 2 } + 2 x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes.
  3. Find the value of the discriminant of \(x ^ { 2 } + 2 x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). The equation \(x ^ { 2 } + k x + 3 = 0\), where \(k\) is a constant, has no real roots.
  4. Find the set of possible values of \(k\), giving your answer in surd form.
Edexcel C1 2010 January Q10
10 marks Moderate -0.3
10. $$\mathrm { f } ( x ) = x ^ { 2 } + 4 k x + ( 3 + 11 k ) , \quad \text { where } k \text { is a constant. }$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found in terms of \(k\). Given that the equation \(\mathrm { f } ( x ) = 0\) has no real roots,
  2. find the set of possible values of \(k\). Given that \(k = 1\),
  3. sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any point at which the graph crosses a coordinate axis.
OCR C1 Specimen Q3
6 marks Moderate -0.8
3 The quadratic equation \(x ^ { 2 } + k x + k = 0\) has no real roots for \(x\).
  1. Write down the discriminant of \(x ^ { 2 } + k x + k\) in terms of \(k\).
  2. Hence find the set of values that \(k\) can take.
OCR MEI C1 2009 January Q8
4 marks Moderate -0.5
8 Find the range of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 18 = 0\) does not have real roots.
OCR MEI C1 2008 June Q3
5 marks Moderate -0.8
3
  1. Solve the equation \(2 x ^ { 2 } + 3 x = 0\).
  2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 x - k = 0\) has no real roots.
OCR PURE Q1
3 marks Moderate -0.3
1 The quadratic equation \(k x ^ { 2 } + 3 x + k = 0\) has no real roots. Determine the set of possible values of \(k\).
AQA AS Paper 2 2022 June Q4
4 marks Moderate -0.8
4 The equation \(9 x ^ { 2 } + 4 x + p ^ { 2 } = 0\) has no real solutions for \(x\). Find the set of possible values of \(p\).
Fully justify your answer.
[0pt] [4 marks]
CAIE P3 2018 June Q2
5 marks Moderate -0.8
The equation of a curve is \(y = x^2 - 6x + k\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis. [2]
  2. Find the value of \(k\) for which the line \(y + 2x = 7\) is a tangent to the curve. [3]
Edexcel C1 Q5
4 marks Moderate -0.3
The equation \(2x^2 - 3x - (k + 1) = 0\), where \(k\) is a constant, has no real roots. Find the set of possible values of \(k\). [4]
Edexcel C1 Q4
7 marks Moderate -0.8
\(f(x) = x^2 - kx + 9\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the equation \(f(x) = 0\) has no real solutions. [4]
Given that \(k = 4\),
  1. express \(f(x)\) in the form \((x - p)^2 + q\), where \(p\) and \(q\) are constants to be found, [3]
OCR C1 2013 January Q8
7 marks Standard +0.3
The quadratic equation \(kx^2 + (3k - 1)x - 4 = 0\) has no real roots. Find the set of possible values of \(k\). [7]
OCR MEI C1 2006 January Q9
5 marks Moderate -0.8
  1. Find the range of values of \(k\) for which the equation \(x^2 + 5x + k = 0\) has one or more real roots. [3]
  2. Solve the equation \(4x^2 + 20x + 25 = 0\). [2]
OCR MEI C1 2012 June Q7
4 marks Moderate -0.5
Find the set of values of \(k\) for which the graph of \(y = x^2 + 2kx + 5\) does not intersect the \(x\)-axis. [4]
OCR MEI C1 Q12
4 marks Standard +0.3
Find the range of values of \(k\) for which the equation \(2x^2 + kx + 18 = 0\) does not have real roots. [4]
OCR MEI C1 Q1
5 marks Moderate -0.8
  1. Solve the equation \(2x^2 + 3x = 0\). [2]
  2. Find the set of values of \(k\) for which the equation \(2x^2 + 3x - k = 0\) has no real roots. [3]
OCR MEI C1 Q7
4 marks Standard +0.3
Find the set of values of \(k\) for which the equation \(2x^2 + kx + 2 = 0\) has no real roots. [4]
OCR MEI C1 Q11
5 marks Moderate -0.8
  1. Find the range of values of \(k\) for which the equation \(x^2 + 5x + k = 0\) has one or more real roots. [3]
  2. Solve the equation \(4x^2 + 20x + 25 = 0\). [2]
OCR MEI AS Paper 2 2018 June Q5
3 marks Moderate -0.8
Find the set of values of \(a\) for which the equation $$ax^2 + 8x + 2 = 0$$ has no real roots. [3]
WJEC Unit 1 2022 June Q9
4 marks Standard +0.3
Find the range of values of \(k\) for which the quadratic equation \(x^2 + 2kx + 8k = 0\) has no real roots. [4]
SPS SPS SM 2024 October Q4
7 marks Standard +0.3
The quadratic equation \(kx^2 + (3k - 1)x - 4 = 0\) has no real roots. Find the set of possible values of \(k\). [7]