4.05b Transform equations: substitution for new roots

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

1. determine the value of $$\left( \alpha ^ { 2 } + 1 \right) \left( \beta ^ { 2 } + 1 \right)$$
2. 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.

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

1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
2. determine, giving each answer as a simplified fraction, the value of
   1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
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.

1. The quadratic equation

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

1. find the exact value of
   1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
2. 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.

   VIAV SIHI NI BIIIM ION OC

   VGHV SIHI NI GHIYM ION OC

   VJ4V SIHI NI JIIYM ION OC

1. The quadratic equation

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

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

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

1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
2. find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
3. 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.

4 Use the substitution \(x = u + 2\) to find the exact value of the real root of the equation $$x ^ { 3 } - 6 x ^ { 2 } + 12 x - 13 = 0$$

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

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

3 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 24 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = \frac { 1 } { u }\) to find a cubic equation in \(u\) with integer coefficients.
2. Hence, or otherwise, find the value of \(\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }\).

10 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } + 3 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\). The cubic equation \(x ^ { 3 } + p x ^ { 2 } + 10 x + q = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).
2. Find the value of \(p\).
3. Find the value of \(q\).

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 \(\frac { \beta } { \alpha }\).

5 The roots of the cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), simplifying your answer as far as you can.

5 The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. Find the cubic equation with roots \(2 \alpha , 2 \beta\) and \(2 \gamma\), simplifying your answer as far as possible.

5 The roots of the cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(2 \alpha + 1,2 \beta + 1\) and \(2 \gamma + 1\), expressing your answer in a form with integer coefficients.

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

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots \(2 \alpha\) and \(2 \beta\).

6 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } - 3 x + 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(2 \alpha , 2 \beta\) and \(2 \gamma\), expressing your answer in a form with integer coefficients.

8

1. 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 \(\alpha > \beta\).
2. Show that \(\alpha - \beta = 4 \sqrt { 2 } k\).
3. Hence find a quadratic equation with roots \(\alpha + 1\) and \(\beta - 1\).

2 The cubic equation \(2 x ^ { 3 } + 3 x - 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = u - 1\) to find a cubic equation in \(u\) with integer coefficients.
2. Hence find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).

5 The cubic equation \(x ^ { 3 } + 5 x ^ { 2 } + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = \sqrt { u }\) to find a cubic equation in \(u\) with integer coefficients.
2. Hence find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).

10 The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = \frac { 1 } { \sqrt { u } }\) to show that \(4 u ^ { 3 } + 12 u ^ { 2 } + 9 u - 1 = 0\).
2. Hence find the values of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) and \(\frac { 1 } { \alpha ^ { 2 } \beta ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } \alpha ^ { 2 } }\).

6 The quadratic equation \(2 x ^ { 2 } + x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Use the substitution \(x = \frac { 1 } { u + 1 }\) to obtain a quadratic equation in \(u\) with integer coefficients.
2. Hence, or otherwise, find the value of \(\left( \frac { 1 } { \alpha } - 1 \right) \left( \frac { 1 } { \beta } - 1 \right)\).

5 The cubic equation \(2 x ^ { 3 } + 3 x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = u + 2\) to find a cubic equation in \(u\).
2. Hence find the value of \(\frac { 1 } { \alpha - 2 } + \frac { 1 } { \beta - 2 } + \frac { 1 } { \gamma - 2 }\).

10 The cubic equation \(x ^ { 3 } + 4 x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = \sqrt { u }\) to obtain a cubic equation in \(u\).
2. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \alpha \beta \gamma\).

3 The roots of the cubic equation \(x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).

5 The equation \(z ^ { 3 } - 5 z ^ { 2 } + 3 z - 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(\frac { \alpha } { 2 } + 1 , \frac { \beta } { 2 } + 1\), \(\frac { \gamma } { 2 } + 1\), expressing your answer in a form with integer coefficients.