Q.The arithmetic mean and the geometric mean of two positive numbers are $13$ and $12$ respectively. Find the numbers.

[!TLDR] From $A=13,G=12$: $a+b=26$, $ab=144$; solve $x^2-26x+144=0$. [!ANSW …

$a+b=2A=26$, $ab=G^2=144$. Quadratic: $x^2-26x+144=0$. Discriminant $=676-576=100$, $\sqrt{100}=10$. $x=\dfrac{26\pm10}{2}=18$ or $8$. Check: $\dfrac{8+18}{2}=1 …

Form $x^2-2Ax+G^2=0$ from the given A.M. and G.M., and solve …

Forgetting to square $G$ when writing the product term (using $ab=G$ instead of $ab=G^ …