if A be the AM and G be the GM between two numbers, show that the numbers are given by A + or -  sqrt (A^2 - G^2).

∴ AM=$\frac{a+b}{2}-\left(1\right)$
GM=$\sqrt{ab} -\left(2\right)$
From (1) and (2), we obtain

a+b= 2A    -(3)

ab = $G^2 -\left(4\right)$

Substituting the value of a and b from (3) and (4) in the identity (a – b)² = (a + b)² – 4ab, we obtain

(a – b)² = $4A^2-4G^2=4(A^2-G^2)$

(a – b)² = 4 (A + G) (A – G)

$(a-b)= 2\sqrt{(A+G)(A-G)} -\left(5\right)$

From (3) and (5), we obtain
$$\begin{gathered} 2a=2A +2\sqrt{(A+G)\left(A-G\right)} \\ a=A+\sqrt{\left(A+G\right)\left(A-G\right)} \end{gathered}$$
Substituting the value of a in (3), we obtain
$b=2A-A-\sqrt{\left(A+G\right)\left(A-G\right)}=A-\sqrt{\left(A+G\right)\left(A-G\right)}$
Thus, the two numbers are $A \pm \sqrt{(A+G)(A-G)}$.

Regards