If A be the A.M. and G be the G.M. between two numbers . Show that the numbers are given by $A \pm \sqrt{A^{2} - G^{2}}$

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Solution

Let the $2$ nos. be $a, b$
$A = \frac{a + b}{2} \Rightarrow 2 A = a + b G = \sqrt{a b} \Rightarrow G^{2} = a b$
Quadratic equation having $a, b$ as it roots is
$x^{2} - (a + b) x + a b = 0$
where $(a + b) = 2 A$ and $a b = G^{2}$
$\Rightarrow x^{2} - 2 A x + G^{2} = 0 x = \frac{2 A \pm \sqrt{4 A^{2} - 4 G^{2}}}{2} = \frac{2 A \pm 2 \sqrt{A^{2} - G^{2}}}{2} = A \pm \sqrt{A^{2} - G^{2}}$
Therefore nos. are given by $A \pm \sqrt{A^{2} - G^{2}}$

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