Question

If the arithmetic mean of the roots of the equation x (x – 2) + 4ax = 5 is 3, then a = ________.

Answer 1

Let the roots of the quadratic equation x (x – 2) + 4ax = 5 be $\alpha \mathrm{and}\beta$.

$$\begin{gathered} x\left(x-2\right)+4ax=5 \\ \Rightarrow x^2-2x+4ax-5=0 \\ \Rightarrow x^2+x\left(4a-2\right)-5=0 \\ \mathrm{Sum}\mathrm{of}\mathrm{the}\mathrm{roots}=-\frac{4a-2}{1} \\ \Rightarrow \alpha +\beta =2-4a \\ \mathrm{Arithmetic}\mathrm{Mean}\mathrm{of}\mathrm{the}\mathrm{roots}=3 \\ \Rightarrow \frac{\alpha +\beta }{2}=3 \\ \Rightarrow \frac{2-4a}{2}=3 \\ \Rightarrow 2-4a=6 \\ \Rightarrow -4a=6-2 \\ \Rightarrow -4a=4 \\ \Rightarrow a=-1 \end{gathered}$$

Hence, a = –1.