Show discriminant inequality, then solve

A question is this type if and only if it has a two-part structure where part (a) requires showing that a condition on a constant leads to a specific quadratic inequality in that constant, and part (b) requires solving that inequality to find the range of values.

24 questions · Moderate -0.3

1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable
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Edexcel P1 2023 October Q6
6 marks Standard +0.3
  1. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.} The equation $$4 ( p - 2 x ) = \frac { 12 + 15 p } { x + p } \quad x \neq - p$$ where \(p\) is a constant, has two distinct real roots.
  1. Show that $$3 p ^ { 2 } - 10 p - 8 > 0$$
  2. Hence, using algebra, find the range of possible values of \(p\)
Edexcel C12 2016 January Q13
8 marks Standard +0.3
13. The equation \(k \left( 3 x ^ { 2 } + 8 x + 9 \right) = 2 - 6 x\), where \(k\) is a real constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$11 k ^ { 2 } - 30 k - 9 > 0$$
  2. Find the range of possible values for \(k\).
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Edexcel C12 2018 January Q4
7 marks Moderate -0.3
4. The equation \(( p - 2 ) x ^ { 2 } + 8 x + ( p + 4 ) = 0 , \quad\) where \(p\) is a constant has no real roots.
  1. Show that \(p\) satisfies \(p ^ { 2 } + 2 p - 24 > 0\)
  2. Hence find the set of possible values of \(p\).
Edexcel C12 2014 June Q10
7 marks Moderate -0.3
10. The equation $$k x ^ { 2 } + 4 x + k = 2 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 4 < 0$$
  2. Hence find the set of all possible values of \(k\).
Edexcel C12 2015 June Q9
7 marks Moderate -0.3
9. The equation \(x ^ { 2 } + ( 6 k + 4 ) x + 3 = 0\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$9 k ^ { 2 } + 12 k + 1 < 0$$
  2. Find the range of possible values for \(k\), giving your boundaries as fully simplified surds.
Edexcel C12 2018 June Q8
7 marks Moderate -0.3
8. The equation \(( k - 4 ) x ^ { 2 } - 4 x + k - 2 = 0\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$k ^ { 2 } - 6 k + 4 > 0$$
  2. Find the exact range of possible values for \(k\).
Edexcel C12 2019 June Q11
7 marks Moderate -0.3
11. The straight line \(l\) has equation \(y = m x - 2\), where \(m\) is a constant. The curve \(C\) has equation \(y = 2 x ^ { 2 } + x + 6\) The line \(l\) does not cross or touch the curve \(C\).
  1. Show that \(m\) satisfies the inequality $$m ^ { 2 } - 2 m - 63 < 0$$
  2. Hence find the range of possible values of \(m\).
Edexcel C12 2016 October Q11
8 marks Standard +0.3
11. The equation \(5 x ^ { 2 } + 6 = k \left( 13 x ^ { 2 } - 12 x \right)\), where \(k\) is a constant, has two distinct real roots.
  1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } + 13 k - 5 > 0$$
  2. Find the set of possible values for \(k\).
Edexcel C12 2018 October Q11
8 marks Moderate -0.3
11. The equation \(7 x ^ { 2 } + 2 k x + k ^ { 2 } = k + 7\), where \(k\) is a constant, has two distinct real roots.
  1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } - 7 k - 49 < 0$$
  2. Find the range of possible values for \(k\).
Edexcel C12 Specimen Q8
7 marks Moderate -0.5
8. 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 \(k\).
AQA C1 2005 January Q7
10 marks Standard +0.3
7
  1. Simplify \(( k + 5 ) ^ { 2 } - 12 k ( k + 2 )\).
  2. The quadratic equation \(3 ( k + 2 ) x ^ { 2 } + ( k + 5 ) x + k = 0\) has real roots.
    1. Show that \(( k - 1 ) ( 11 k + 25 ) \leqslant 0\).
    2. Hence find the possible values of \(k\).
AQA C1 2013 January Q8
8 marks Moderate -0.3
8 A curve has equation \(y = 2 x ^ { 2 } - x - 1\) and a line has equation \(y = k ( 2 x - 3 )\), where \(k\) is a constant.
  1. Show that the \(x\)-coordinate of any point of intersection of the curve and the line satisfies the equation $$2 x ^ { 2 } - ( 2 k + 1 ) x + 3 k - 1 = 0$$
  2. The curve and the line intersect at two distinct points.
    1. Show that \(4 k ^ { 2 } - 20 k + 9 > 0\).
    2. Find the possible values of \(k\).
AQA C1 2010 June Q7
12 marks Moderate -0.3
7
    1. Express \(2 x ^ { 2 } - 20 x + 53\) in the form \(2 ( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Use your result from part (a)(i) to explain why the equation \(2 x ^ { 2 } - 20 x + 53 = 0\) has no real roots.
  2. The quadratic equation \(( 2 k - 1 ) x ^ { 2 } + ( k + 1 ) x + k = 0\) has real roots.
    1. Show that \(7 k ^ { 2 } - 6 k - 1 \leqslant 0\).
    2. Hence find the possible values of \(k\).
AQA C1 2013 June Q7
8 marks Moderate -0.3
7 The quadratic equation $$( 2 k - 7 ) x ^ { 2 } - ( k - 2 ) x + ( k - 3 ) = 0$$ has real roots.
  1. Show that \(7 k ^ { 2 } - 48 k + 80 \leqslant 0\).
  2. Find the possible values of \(k\).
Edexcel C1 Q2
7 marks Moderate -0.8
2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.
  1. Prove that \(k ( 25 k - 8 ) \geq 0\).
  2. Hence find the set of possible values of \(k\).
  3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.
Edexcel C1 Q5
7 marks Moderate -0.8
5. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.
  1. Prove that \(k ( 25 k - 8 ) \geq 0\).
  2. Hence find the set of possible values of \(k\).
  3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.
Edexcel C1 Q7
8 marks Moderate -0.3
7. Given that the equation $$4 x ^ { 2 } - k x + k - 3 = 0$$ where \(k\) is a constant, has real roots,
  1. show that $$k ^ { 2 } - 16 k + 48 \geq 0 ,$$
  2. find the set of possible values of \(k\),
  3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value.
Edexcel C1 Q6
7 marks Moderate -0.8
The equation \(x^2 + 5kx + 2k = 0\), where \(k\) is a constant, has real roots.
  1. Prove that \(k(25k - 8) \geq 0\). [2]
  2. Hence find the set of possible values of \(k\). [4]
  3. Write down the values of \(k\) for which the equation \(x^2 + 5kx + 2k = 0\) has equal roots. [1]
Edexcel M2 2014 January Q8
7 marks Moderate -0.8
The equation \(2x^2 + 2kx + (k + 2) = 0\), where \(k\) is a constant, has two distinct real roots.
  1. Show that \(k\) satisfies $$k^2 - 2k - 4 > 0$$ [3]
  2. Find the set of possible values of \(k\). [4]
OCR C1 Q5
8 marks Moderate -0.3
Given that the equation $$4x^2 - kx + k - 3 = 0,$$ where \(k\) is a constant, has real roots,
  1. show that $$k^2 - 16k + 48 \geq 0, \quad [2]$$
  2. find the set of possible values of \(k\), [3]
  3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value. [3]
SPS SPS SM 2022 October Q8
8 marks Standard +0.3
The equation \(k(3x^2 + 8x + 9) = 2 - 6x\), where \(k\) is a real constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$11k^2 - 30k - 9 > 0$$ [4]
  2. Find the range of possible values for \(k\). [4]
SPS SPS SM 2022 October Q4
6 marks Moderate -0.3
The equation $$(k + 3)x^2 + 6x + k = 5$$, where \(k\) is a constant, has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k^2 - 2k - 24 < 0$$ [4]
  2. Hence find the set of possible values of \(k\). [2]
SPS SPS FM 2024 October Q2
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]
SPS SPS SM 2024 October Q4
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]