Question

If the Arithmetic mean and Geometric mean of roots of a quadratic equation are 8 and 5 respectively then obtain the quadratic equation.

Answer 1

Let the two roots be $\alpha$ and $\beta$

$\therefore$ $\frac{\alpha +\beta }{2}=8$ ($AM$ of $a$ and $b$ is $\frac{a+b}{2}$) $⟶\left(1\right)$

and $\sqrt{\alpha \beta }=5$ ($GM$ of $a$ and $b$ is $\sqrt{ab}$) $⟶\left(2\right)$

From $\left(1\right)$

$\alpha +\beta =16$

$\Rightarrow$ sum of roots $=\alpha +\beta =16$

From $\left(2\right)$

$\sqrt{\alpha \beta }=5$

$\Rightarrow \alpha \beta =25$

$\therefore$ Products of roots $=25$

Now, we know that if we have the sum of roots and product of roots then the quadratic equation is given by

$x^2-$ (sum of the roots)$x$ $+$ (product of the roots) $=0$

$\therefore$ the quadratic equation with roots $\alpha ,\beta$ are

$x^2-(\alpha +\beta )x+\alpha \beta =0$

$x^2-16x+25=0$