Construct a quadratic in x such that A.M.
Question:

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

Solution:

Let the roots of the quadratic equation be $a$ and $b$.

$A=\frac{a+b}{2}$

$\therefore a+b=2 A \quad \ldots \ldots \ldots(\mathrm{i})$

Also, $G^{2}=a b \quad \ldots \ldots$ (ii)

The quadratic equation having roots a and b is given by $x^{2}-(a+b) x+a b=0$.

$\therefore x^{2}-2 A x+G^{2}=0 \quad[$ Using (i) and (ii) $]$

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