Write the quadratic equation, the arithmetic and geometric means of whose

Question:

Write the quadratic equation, the arithmetic and geometric means of whose roots are A and G respectively. 

Solution:

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

The arithmetic and geometric means of roots are A and G respectively.

$\Rightarrow A=(a+b) / 2 \ldots(i)$

And $\mathrm{G}=\sqrt{\mathrm{ab}}$ …(ii)

We know that the equation whose roots are given is =

$x^{2}-(a+b) x+a b=0$

From (i) and (ii) we get:

$\mathrm{x}^{2}-2 \mathrm{~A}+\mathrm{G}^{2}=0$

Thus, $x^{2}-2 A+G^{2}=0$ is the required quadratic equation.

Ans: $x^{2}-2 A+G^{2}=0$ is the required quadratic equation.

 

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