Q.The arithmetic mean of two positive numbers is $10$ and their geometric mean is $8$. Find the numbers.

[!TLDR] From $A=10,G=8$: $a+b=20$, $ab=64$; solve $x^2-20x+64=0$. [!ANSWE …

Given $A=10$ and $G=8$, we have $a+b=2A=20$ and $ab=G^2=64$. So $a,b$ are the roots of $x^2-(a+b)x+ab=0$, i.e. $x^2-20x+64=0$. By the quadratic formula, $x=\dfrac{20\pm\sqrt{400-256}}{2}=\dfrac{20\pm\sqrt{144}}{2}=\dfrac{20\pm12}{2}$, giving $x=16$ or $x=4$. Check: $\dfrac{4+16}{2}=10=A$ ✓ and …

Set $a+b=2A$ and $ab=G^2$, form the quadratic $x^2-2Ax+G^2=0$, and sol …

A common error is setting $ab=G$ instead of $ab=G^2$ (forgetting the mean was defined via $G^2=ab$), which …