The equation ${x^2} - 5x - 7 = 0$ has roots $\alpha$ and $\beta$. The equation ${x^2} + px + q = 0$ has roots $\alpha  + 1$ and $\beta  + 1$. Find the value of $p$ and the value of $q$.

METHOD 1

$\alpha  + \beta  = 5,\,\,\alpha \beta  =  - 7$     (M1)(A1)

Note: Award M1A0 if only one equation obtained.

$\left( {\alpha  + 1} \right) + \left( {\beta  + 1} \right) = 5 + 2 = 7$     A1

$\left( {\alpha  + 1} \right)\left( {\beta  + 1} \right) = \alpha \beta  + \left( {\alpha  + \beta } \right) + 1$     (M1)

$=  - 7 + 5 + 1 =  - 1$

$p =  - 7,\,\,q =  - 1$       A1A1

METHOD 2

$\alpha  = \frac{{5 + \sqrt {53} }}{2} = 6.1 \ldots {\text{;}}\,\,\beta  = \frac{{5 - \sqrt {53} }}{2} =  - 1.1 \ldots$     (M1)(A1)

$\alpha  + 1 = \frac{{7 + \sqrt {53} }}{2} = 7.1 \ldots {\text{;}}\,\,\beta  + 1 = \frac{{7 - \sqrt {53} }}{2} =  - 0.1 \ldots$     A1

$\left( {x - 7.14 \ldots } \right)\left( {x + 0.14 \ldots } \right) = {x^2} - 7x - 1$     (M1)

$p =  - 7,\,\,q =  - 1$       A1A1

Note: Exact answers only.

[6 marks]