Quadratic with transformed roots - Roots of polynomials

Edexcel F1 2014 January Q2

8 marks Moderate -0.3

2. The quadratic equation $$5 x ^ { 2 } - 4 x + 2 = 0$$ has roots $\alpha$ and $\beta$.

Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.
Find the value of $\alpha ^ { 2 } + \beta ^ { 2 }$.
Find a quadratic equation which has roots $$\frac { 1 } { \alpha ^ { 2 } } \text { and } \frac { 1 } { \beta ^ { 2 } }$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers.
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Edexcel F1 2015 January Q5

8 marks Standard +0.3

5. The quadratic equation $$4 x ^ { 2 } + 3 x + 1 = 0$$ has roots $\alpha$ and $\beta$.

Write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$.
Find the value of $\left( \alpha ^ { 2 } + \beta ^ { 2 } \right)$.
Find a quadratic equation which has roots $$( 4 \alpha - \beta ) \text { and } ( 4 \beta - \alpha )$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.

Edexcel F1 2016 January Q3

11 marks Standard +0.8

3. The quadratic equation $$x ^ { 2 } - 2 x + 3 = 0$$ has roots $\alpha$ and $\beta$.
Without solving the equation,

1. write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
2. show that $\alpha ^ { 2 } + \beta ^ { 2 } = - 2$
3. find the value of $\alpha ^ { 3 } + \beta ^ { 3 }$
1. show that $\alpha ^ { 4 } + \beta ^ { 4 } = \left( \alpha ^ { 2 } + \beta ^ { 2 } \right) ^ { 2 } - 2 ( \alpha \beta ) ^ { 2 }$
2. find a quadratic equation which has roots $$\text { ( } \alpha ^ { 3 } - \beta \text { ) and ( } \beta ^ { 3 } - \alpha \text { ) }$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers.

Edexcel F1 2018 January Q4

8 marks Standard +0.8

The quadratic equation

$$3 x ^ { 2 } + 2 x + 5 = 0$$ has roots $\alpha$ and $\beta$. Without solving the equation,

find the value of $\alpha ^ { 2 } + \beta ^ { 2 }$
show that $\alpha ^ { 3 } + \beta ^ { 3 } = \frac { 82 } { 27 }$
find a quadratic equation which has roots $$\left( \alpha + \frac { \alpha } { \beta ^ { 2 } } \right) \text { and } \left( \beta + \frac { \beta } { \alpha ^ { 2 } } \right)$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers.

Edexcel F1 2021 January Q4

8 marks Standard +0.8

The equation $2 x ^ { 2 } + 5 x + 7 = 0$ has roots $\alpha$ and $\beta$

Without solving the equation

determine the exact value of $\alpha ^ { 3 } + \beta ^ { 3 }$
form a quadratic equation, with integer coefficients, which has roots $$\frac { \alpha ^ { 2 } } { \beta } \text { and } \frac { \beta ^ { 2 } } { \alpha }$$ \includegraphics[max width=\textwidth, alt={}, center]{f8660b02-384e-460f-a0e4-282ef5fef475-09_2255_50_314_34}

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Edexcel F1 2022 January Q6

8 marks Standard +0.8

6. The quadratic equation $$A x ^ { 2 } + 5 x - 12 = 0$$ where $A$ is a constant, has roots $\alpha$ and $\beta$

Write down an expression in terms of $A$ for
1. $\alpha + \beta$
2. $\alpha \beta$ The equation $$4 x ^ { 2 } - 5 x + B = 0$$ where $B$ is a constant, has roots $\alpha - \frac { 3 } { \beta }$ and $\beta - \frac { 3 } { \alpha }$
Determine the value of $A$
Determine the value of $B$ The rectangular hyperbola $H$ has equation $x y = 36$ The point $P ( 4,9 )$ lies on $H$
Show, using calculus, that the normal to $H$ at $P$ has equation $$4 x - 9 y + 65 = 0$$ The normal to $H$ at $P$ crosses $H$ again at the point $Q$
Determine an equation for the tangent to $H$ at $Q$, giving your answer in the form $y = m x + c$ where $m$ and $c$ are rational constants. \section*{7. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
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Edexcel F1 2023 January Q5

9 marks Standard +0.8

The quadratic equation

$$4 x ^ { 2 } + 3 x + k = 0$$ where $k$ is an integer, has roots $\alpha$ and $\beta$

Write down, in terms of $k$ where appropriate, the value of $\alpha + \beta$ and the value of $\alpha \beta$
Determine, in simplest form in terms of $k$, the value of $\frac { \alpha } { \beta ^ { 2 } } + \frac { \beta } { \alpha ^ { 2 } }$
Determine a quadratic equation which has roots $$\frac { \alpha } { \beta ^ { 2 } } \text { and } \frac { \beta } { \alpha ^ { 2 } }$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integer values in terms of $k$

Edexcel F1 2024 January Q5

9 marks Standard +0.8

The quadratic equation

$$2 x ^ { 2 } - 3 x + 7 = 0$$ has roots $\alpha$ and $\beta$
Without solving the equation,

write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
determine the value of $\alpha ^ { 2 } + \beta ^ { 2 }$
find a quadratic equation which has roots $$\left( \alpha - \frac { 1 } { \beta ^ { 2 } } \right) \text { and } \left( \beta - \frac { 1 } { \alpha ^ { 2 } } \right)$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.

Edexcel F1 2014 June Q6

8 marks Standard +0.8

6. It is given that $\alpha$ and $\beta$ are roots of the equation $3 x ^ { 2 } + 5 x - 1 = 0$

Find the exact value of $\alpha ^ { 3 } + \beta ^ { 3 }$
Find a quadratic equation which has roots $\frac { \alpha ^ { 2 } } { \beta }$ and $\frac { \beta ^ { 2 } } { \alpha }$, giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a$, $b$ and $c$ are integers.

Edexcel F1 2015 June Q3

6 marks Standard +0.3

3. It is given that $\alpha$ and $\beta$ are roots of the equation $$2 x ^ { 2 } - 7 x + 4 = 0$$

Find the exact value of $\alpha ^ { 2 } + \beta ^ { 2 }$
Find a quadratic equation which has roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$, giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are integers.

Edexcel F1 2016 June Q9

9 marks Standard +0.8

9. The quadratic equation $$2 x ^ { 2 } + 4 x - 3 = 0$$ has roots $\alpha$ and $\beta$.
Without solving the quadratic equation,

find the exact value of
1. $\alpha ^ { 2 } + \beta ^ { 2 }$
2. $\alpha ^ { 3 } + \beta ^ { 3 }$
Find a quadratic equation which has roots ( $\alpha ^ { 2 } + \beta$ ) and ( $\beta ^ { 2 } + \alpha$ ), giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are integers.
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Edexcel F1 2018 June Q7

9 marks Standard +0.3

7. It is given that $\alpha$ and $\beta$ are roots of the equation $5 x ^ { 2 } - 4 x + 3 = 0$ Without solving the quadratic equation,

find the exact value of $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }$
find a quadratic equation which has roots $\frac { 3 } { \alpha ^ { 2 } }$ and $\frac { 3 } { \beta ^ { 2 } }$
giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a$, $b$ and $c$ are integers to be found.

Edexcel F1 2020 June Q2

9 marks Standard +0.3

2
2. The quadratic equation $$5 x ^ { 2 } - 2 x + 3 = 0$$ has roots $\alpha$ and $\beta$.
Without solving the equation,

write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
determine, giving each answer as a simplified fraction, the value of
1. $\alpha ^ { 2 } + \beta ^ { 2 }$
2. $\alpha ^ { 3 } + \beta ^ { 3 }$
determine a quadratic equation that has roots $$\left( \alpha + \beta ^ { 2 } \right) \text { and } \left( \beta + \alpha ^ { 2 } \right)$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers.

Edexcel F1 2022 June Q5

10 marks Standard +0.8

The quadratic equation

$$2 x ^ { 2 } - 3 x + 5 = 0$$ has roots $\alpha$ and $\beta$
Without solving the equation,

write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
determine the value of
1. $\alpha ^ { 2 } + \beta ^ { 2 }$
2. $\alpha ^ { 3 } + \beta ^ { 3 }$
find a quadratic equation which has roots $$\left( \alpha ^ { 3 } - \beta \right) \text { and } \left( \beta ^ { 3 } - \alpha \right)$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.

Edexcel F1 2023 June Q5

10 marks Standard +0.8

5. $$f ( x ) = x ^ { 2 } - 6 x + 3$$ The equation $\mathrm { f } ( x ) = 0$ has roots $\alpha$ and $\beta$
Without solving the equation,

determine the value of $$\left( \alpha ^ { 2 } + 1 \right) \left( \beta ^ { 2 } + 1 \right)$$
find a quadratic equation which has roots $$\frac { \alpha } { \left( \alpha ^ { 2 } + 1 \right) } \text { and } \frac { \beta } { \left( \beta ^ { 2 } + 1 \right) }$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.

Edexcel F1 2024 June Q5

9 marks Challenging +1.2

The equation $5 x ^ { 2 } - 4 x + 2 = 0$ has roots $\frac { 1 } { p }$ and $\frac { 1 } { q }$
1. Without solving the equation,
  1. show that $p q = \frac { 5 } { 2 }$
  2. determine the value of $p + q$
2. Hence, without finding the values of $p$ and $q$, determine a quadratic equation with roots
$$\frac { p } { p ^ { 2 } + 1 } \text { and } \frac { q } { q ^ { 2 } + 1 }$$ giving your answer in the form $a x ^ { 2 } + b x + c = 0$ where $a , b$ and $c$ are integers.

Edexcel F1 2021 October Q3

9 marks Standard +0.8

3. The quadratic equation $$2 x ^ { 2 } - 5 x + 7 = 0$$ has roots $\alpha$ and $\beta$ Without solving the equation,

write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
determine, giving each answer as a simplified fraction, the value of
1. $\alpha ^ { 2 } + \beta ^ { 2 }$
2. $\alpha ^ { 3 } + \beta ^ { 3 }$
find a quadratic equation that has roots $$\frac { 1 } { \alpha ^ { 2 } + \beta } \text { and } \frac { 1 } { \beta ^ { 2 } + \alpha }$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.

Edexcel F1 2018 Specimen Q9

9 marks Standard +0.8

The quadratic equation

$$2 x ^ { 2 } + 4 x - 3 = 0$$ has roots $\alpha$ and $\beta$.
Without solving the quadratic equation,

find the exact value of
1. $\alpha ^ { 2 } + \beta ^ { 2 }$
2. $\alpha ^ { 3 } + \beta ^ { 3 }$
Find a quadratic equation which has roots ( $\alpha ^ { 2 } + \beta$ ) and ( $\beta ^ { 2 } + \alpha$ ), giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are integers.
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Edexcel F1 Specimen Q4

12 marks Standard +0.3

The quadratic equation

$$5 x ^ { 2 } - 4 x + 1 = 0$$ has roots $\alpha$ and $\beta$.

Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.
Show that $\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 6 } { 5 }$
Find a quadratic equation with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \text { and } \beta + \frac { 1 } { \beta }$$

Edexcel F1 2017 January Q2

7 marks Standard +0.3

The quadratic equation $$2 x ^ { 2 } - x + 3 = 0$$ has roots $\alpha$ and $\beta$.
Without solving the equation,

write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
find the value of $\frac { 1 } { \alpha } + \frac { 1 } { \beta }$
find a quadratic equation which has roots $$\left( 2 \alpha - \frac { 1 } { \beta } \right) \text { and } \left( 2 \beta - \frac { 1 } { \alpha } \right)$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers.

OCR FP1 2007 January Q7

8 marks Standard +0.3

7 The quadratic equation $x ^ { 2 } + 5 x + 10 = 0$ has roots $\alpha$ and $\beta$.

Write down the values of $\alpha + \beta$ and $\alpha \beta$.
Show that $\alpha ^ { 2 } + \beta ^ { 2 } = 5$.
Hence find a quadratic equation which has roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$.

OCR FP1 2008 June Q8

7 marks Standard +0.3

8 The quadratic equation $x ^ { 2 } + k x + 2 k = 0$, where $k$ is a non-zero constant, has roots $\alpha$ and $\beta$. Find a quadratic equation with roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$.

OCR FP1 2009 January Q8

10 marks Standard +0.3

8

Show that $( \alpha - \beta ) ^ { 2 } \equiv ( \alpha + \beta ) ^ { 2 } - 4 \alpha \beta$. The quadratic equation $x ^ { 2 } - 6 k x + k ^ { 2 } = 0$, where $k$ is a positive constant, has roots $\alpha$ and $\beta$, with $\alpha > \beta$.
Show that $\alpha - \beta = 4 \sqrt { 2 } k$.
Hence find a quadratic equation with roots $\alpha + 1$ and $\beta - 1$.

OCR FP1 2011 January Q8

9 marks Standard +0.3

8 The quadratic equation $2 x ^ { 2 } - x + 3 = 0$ has roots $\alpha$ and $\beta$, and the quadratic equation $x ^ { 2 } - p x + q = 0$ has roots $\alpha + \frac { 1 } { \alpha }$ and $\beta + \frac { 1 } { \beta }$.

Show that $p = \frac { 5 } { 6 }$.
Find the value of $q$.

OCR FP1 2010 June Q7

7 marks Standard +0.8

7 The quadratic equation $x ^ { 2 } + 2 k x + k = 0$, where $k$ is a non-zero constant, has roots $\alpha$ and $\beta$. Find a quadratic equation with roots $\frac { \alpha + \beta } { \alpha }$ and $\frac { \alpha + \beta } { \beta }$.

Show that $\frac { 1 } { \sqrt { r + 2 } + \sqrt { r } } \equiv \frac { \sqrt { r + 2 } - \sqrt { r } } { 2 }$.
Hence find an expression, in terms of $n$, for $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$$
State, giving a brief reason, whether the series $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$ converges.