Question

If the difference between the roots of the equation $x^2+ax+8=0$ is 2, write the values of a.

Answer 1

Given: $x^2+ax+8=0.$
Let $\alpha and\beta$ are the roots of the equation.
Sum of the roots = $\alpha +\beta =\frac{-a}{1}=-a$.

Product of the roots = $\alpha \beta =\frac{8}{1}=8$
Given: $\alpha -\beta =2$

$$\begin{gathered} \mathrm{Then,}{\left(\alpha +\beta \right)}^2-{\left(\alpha -\beta \right)}^2=4\alpha \beta \\ \Rightarrow {\left(\alpha +\beta \right)}^2-2^2=4\times 8 \\ \Rightarrow {\left(\alpha +\beta \right)}^2-4=32 \\ \Rightarrow {\left(\alpha +\beta \right)}^2=32+4=36 \\ \Rightarrow \left(\alpha +\beta \right)=\pm 6 \end{gathered}$$
$\alpha +\beta =-a=\pm 6$
∴ $a=\pm 6$