Quadratic with transformed roots

A question is this type if and only if it gives a quadratic equation with roots α, β and asks to find a new quadratic with roots that are specific expressions involving α and β (like α/β + β/α, or α² + β).

48 questions · Standard +0.5

4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots
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AQA FP1 2011 January Q1
7 marks Standard +0.3
1 The quadratic equation \(x ^ { 2 } - 6 x + 18 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find a quadratic equation, with integer coefficients, which has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
  3. Hence find the values of \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
AQA FP1 2013 January Q5
9 marks Standard +0.8
5 The roots of the quadratic equation $$x ^ { 2 } + 2 x - 5 = 0$$ are \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Calculate the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots \(\alpha ^ { 3 } \beta + 1\) and \(\alpha \beta ^ { 3 } + 1\).
AQA FP1 2007 June Q4
7 marks Standard +0.3
4 The quadratic equation $$2 x ^ { 2 } - x + 4 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 1 } { 4 }\).
  3. Find a quadratic equation with integer coefficients such that the roots of the equation are $$\frac { 4 } { \alpha } \text { and } \frac { 4 } { \beta }$$ (3 marks)
AQA FP1 2008 June Q1
8 marks Standard +0.3
1 The equation $$x ^ { 2 } + x + 5 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = - \frac { 9 } { 5 }\).
  4. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).
AQA FP1 2009 June Q1
7 marks Standard +0.3
1 The equation $$2 x ^ { 2 } + x - 8 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots \(4 \alpha ^ { 2 }\) and \(4 \beta ^ { 2 }\). Give your answer in the form \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are integers.
AQA FP1 2010 June Q8
10 marks Standard +0.8
8 The quadratic equation $$x ^ { 2 } - 4 x + 10 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 2 } { 5 }\).
  3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha + \frac { 2 } { \beta }\) and \(\beta + \frac { 2 } { \alpha }\).
AQA FP1 2011 June Q2
9 marks Standard +0.8
2 The equation $$4 x ^ { 2 } + 6 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = \frac { 3 } { 4 }\).
  3. Find an equation, with integer coefficients, which has roots $$3 \alpha - \beta \text { and } 3 \beta - \alpha$$
AQA FP1 2012 June Q1
10 marks Standard +0.3
1 The quadratic equation $$5 x ^ { 2 } - 7 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 39 } { 5 }\).
  3. Find a quadratic equation, with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \quad \text { and } \quad \beta + \frac { 1 } { \beta }$$ (5 marks)
AQA FP1 2013 June Q6
11 marks Standard +0.8
6 The equation $$2 x ^ { 2 } + 3 x - 6 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Hence show that \(\alpha ^ { 3 } + \beta ^ { 3 } = - \frac { 135 } { 8 }\).
  3. Find a quadratic equation, with integer coefficients, whose roots are \(\alpha + \frac { \alpha } { \beta ^ { 2 } }\) and \(\beta + \frac { \beta } { \alpha ^ { 2 } }\).
AQA FP1 2015 June Q1
9 marks Standard +0.3
1 The quadratic equation \(2 x ^ { 2 } + 6 x + 7 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Find a quadratic equation, with integer coefficients, which has roots \(\alpha ^ { 2 } - 1\) and \(\beta ^ { 2 } - 1\).
  3. Hence find the values of \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
OCR MEI Further Pure Core AS 2019 June Q2
3 marks Standard +0.3
2 The roots of the equation \(3 x ^ { 2 } - x + 2 = 0\) are \(\alpha\) and \(\beta\).
Find a quadratic equation with integer coefficients whose roots are \(2 \alpha - 3\) and \(2 \beta - 3\).
OCR MEI Further Pure Core AS 2021 November Q2
3 marks Standard +0.3
2 The equation \(3 x ^ { 2 } - 4 x + 2 = 0\) has roots \(\alpha\) and \(\beta\).
Find an equation with integer coefficients whose roots are \(3 - 2 \alpha\) and \(3 - 2 \beta\).
AQA FP1 2005 January Q1
7 marks Standard +0.3
1 The equation $$x ^ { 2 } - 5 x - 2 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find the value of \(\alpha ^ { 2 } \beta + \alpha \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots $$\alpha ^ { 2 } \beta \quad \text { and } \quad \alpha \beta ^ { 2 }$$
AQA FP1 2008 January Q8
12 marks Standard +0.8
8
    1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
    2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
  2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
  3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$
AQA FP1 2010 January Q1
9 marks Standard +0.8
1 The quadratic equation $$3 x ^ { 2 } - 6 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 6\).
  3. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\).
AQA FP1 2005 June Q6
11 marks Standard +0.3
6 The equation $$x ^ { 2 } - 4 x + 13 = 0$$ has roots \(\alpha\) and \(\beta\).
    1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
    2. Deduce that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\).
    3. Explain why the statement \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\) implies that \(\alpha\) and \(\beta\) cannot both be real.
  2. Find in the form \(p + \mathrm { i } q\) the values of:
    1. \(( \alpha + \mathrm { i } ) + ( \beta + \mathrm { i } )\);
    2. \(( \alpha + \mathrm { i } ) ( \beta + \mathrm { i } )\).
  3. Hence find a quadratic equation with roots \(( \alpha + \mathrm { i } )\) and \(( \beta + \mathrm { i } )\).
AQA FP1 2006 June Q1
9 marks Standard +0.3
1 The quadratic equation $$3 x ^ { 2 } - 6 x + 2 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the numerical values of \(\alpha + \beta\) and \(\alpha \beta\).
    1. Expand \(( \alpha + \beta ) ^ { 3 }\).
    2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 4\).
  3. Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\), giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.
OCR Further Pure Core 1 2021 June Q1
4 marks Standard +0.3
1 In this question you must show detailed reasoning.
The quadratic equation \(x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Hence find a quadratic equation with roots \(\alpha + \frac { 1 } { \beta }\) and \(\beta + \frac { 1 } { \alpha }\). Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).
AQA FP1 2016 June Q1
7 marks Moderate -0.3
The quadratic equation \(x^2 - 6x + 14 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
  2. Find a quadratic equation, with integer coefficients, which has roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). [5 marks]
OCR FP1 Q8
11 marks Moderate -0.3
  1. The quadratic equation \(x^2 - 2x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
    1. Write down the values of \(\alpha + \beta\) and \(\alpha\beta\). [2]
    2. Show that \(\alpha^2 + \beta^2 = -4\). [2]
    3. Hence find a quadratic equation which has roots \(\alpha^2\) and \(\beta^2\). [3]
  2. The cubic equation \(x^3 - 12x^2 + ax - 48 = 0\) has roots \(p\), \(2p\) and \(3p\).
    1. Find the value of \(p\). [2]
    2. Hence find the value of \(a\). [2]
AQA Further AS Paper 1 2020 June Q9
8 marks Standard +0.3
The quadratic equation \(2x^2 + px + 3 = 0\) has two roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\).
    1. Write down the value of \(\alpha\beta\). [1 mark]
    2. Express \(\alpha + \beta\) in terms of \(p\). [1 mark]
  2. Hence find \((\alpha - \beta)^2\) in terms of \(p\). [2 marks]
  3. Hence find, in terms of \(p\), a quadratic equation with roots \(\alpha - 1\) and \(\beta + 1\) [4 marks]
AQA Further Paper 2 2023 June Q13
11 marks Challenging +1.8
The quadratic equation \(z^2 - 5z + 8 = 0\) has roots \(\alpha\) and \(\beta\)
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\) [2 marks]
  2. Without finding the value of \(\alpha\) or the value of \(\beta\), show that \(\alpha^4 + \beta^4 = -47\) [4 marks]
  3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha^3 + \beta\) and \(\beta^3 + \alpha\) [5 marks]
OCR Further Pure Core 2 2018 September Q7
9 marks Challenging +1.2
The roots of the equation \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are positive integers, are \(\alpha\) and \(\beta\).
  1. Find a quadratic equation with integer coefficients whose roots are \(\alpha + \beta\) and \(\alpha\beta\). [4]
  2. Show that it is not possible for the original equation and the equation found in part (i) both to have repeated roots. [2]
  3. Show that the discriminant of the equation found in part (i) is always positive. [3]