METHOD 1

product of the roots is ${r_1} \times {r_2} \times 1 \times d{\text{i}} \times - d{\text{i}} = - 8{d^2}$ (M1)(A1)

${r_1} \times {r_2} = - 8$ A1

sum of the roots is ${r_1} + {r_2} + 1 + d{\text{i}} + - d{\text{i}} = 3$ (M1)(A1)

${r_1} + {r_2} = 2$ A1

solving simultaneously (M1)

${r_1} = - 2$ , ${r_2} = 4$ A1A1

METHOD 2

product of the roots ${\text{lo}}{{\text{g}}_2}\,a \times {\text{lo}}{{\text{g}}_2}\,b \times {\text{lo}}{{\text{g}}_2}\,c \times d{\text{i}} \times - d{\text{i}} = - 8{d^2}$ M1A1

${\text{lo}}{{\text{g}}_2}\,a \times {\text{lo}}{{\text{g}}_2}\,b \times {\text{lo}}{{\text{g}}_2}\,c = - 8$ A1

EITHER

$a$ , $b$ , $c$ can be written as $\frac{2}{r}$ , $2$ , $2r$ M1

$\left( {{\text{lo}}{{\text{g}}_2}\frac{2}{r}} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2r} \right) = - 8$

attempt to solve M1

$\left( {1 - {\text{lo}}{{\text{g}}_2}\,r} \right)\left( {1 + {\text{lo}}{{\text{g}}_2}\,r} \right) = - 8$

${\text{lo}}{{\text{g}}_2}\,r = \pm 3$

$r = \frac{1}{8}{\text{,}}\,\,8$ A1A1

OR

$a$ , $b$ , $c$ can be written as $a$ , $2$ , $\frac{4}{a}$ M1

$\left( {{\text{lo}}{{\text{g}}_2}\,a} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2} \right)\left( {{\text{lo}}{{\text{g}}_2}\,\frac{4}{a}} \right) = - 8$

attempt to solve M1

$a = \frac{1}{4}{\text{,}}\,\,16$ A1A1

THEN

$a$ and $c$ are $\frac{1}{4}{\text{,}}\,\,16$ (A1)

roots are−2, 4 A1

[9 marks]