Question

If AM and GM of roots of a quadratic equation are 8 and 5 respectively, then obtain the quadratic equation.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{quadratic}\mathrm{equation}\mathrm{be}a\mathrm{and}b\mathit{.} \\ \mathrm{AM}=8 \\ \therefore \frac{a+b}{2}=8 \\ \Rightarrow a+b=16........\left(\mathrm{i}\right) \\ \mathrm{Also},G=5 \\ \Rightarrow \sqrt{ab}=5 \\ \Rightarrow ab=5^2 \\ \Rightarrow ab=25........\left(\mathrm{ii}\right) \\ \mathrm{Now},\mathrm{the}\mathrm{quadratic}\mathrm{equation}\mathrm{is}\mathrm{given}\mathrm{by} \\ {x}^2-(a+b)x+ab=0 \\ \Rightarrow x^2-16x+25=0 \end{gathered}$$