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# If A.M. and G.M. of roots of a quadratic equation are $8$ and $5$ respectively. Find the quadratic equation.

Toolbox:
• A.M. between two numbers $a$ and $b= \large\frac{a+b}{2}$
• G.M. between the two numbers$= \sqrt {ab}$
• If the roots of a quadratic equation are $a$ and $b$ then the equation is $x^2-$ (Sum of the roots)$x+$(product of the roots)=0.
Let the roots of the quadratic equation be $a$ and $b$
It is given that the A.M of the roots $=8$ and their G.M.$=5$
We know that the A.M between $a$ and $b=\large\frac{a+b}{2}$ and
G.M.$=\sqrt {ab}$
$\Rightarrow\:\large\frac{a+b}{2}$$=8$ and $\sqrt {ab}=5$
$\Rightarrow\:a+b=16$ and $ab=25$
Since $a$ and $b$ are roots of the quadratic equation,
sum of the roots $=a+b=16$
Product of the roots $=ab=25$
We know that the general quadratic equation is
$x^2-$(Sum of the roots)$x+$(product of the roots)=0.
$i.e.,$ The required quadratic equation is $x^2-16x+25=0$