If $A.M.$ and $G.M.$ of roots of a quadratic equation are $8$ and $5,$ respectively, then obtain the quadratic equation.
If $A.M.$ and $G.M.$ of roots of a quadratic equation are $8$ and $5,$ respectively, then obtain the quadratic equation.
Let the root of the quadratic equation be $a$ and $b$
According to the given condition,
$A. M.=\frac{a+b}{2}=8 \Rightarrow a+b=16$ ........$(1)$
$G.M.$ $=\sqrt{a b}=5 \Rightarrow a b=25$ .........$(2)$
The quadratic equation is given by,
$x^{2}-x(\text { Sumof roots })+(\text { Product of roots })=0$
$x^{2}-x(a+b)+(a b)=0$
$x^{2}-16 x+25=0$ [ Using $(1)$ and $(2)$ ]
Thus, the required quadratic equation is $x^{2}-16 x+25=0$
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