Alg8.15

If the Arithmetic mean(A.M) and Geometric Mean (G.M) of two numbers are $13,12\,$ respectively.Find the numbers.

Let the two numbers be $x,y\,$

The AM of the two numbers is $\frac{x+y}{2}=13,x+y=26\,$

The GM of the numbers is $\sqrt{xy}=12,xy=144\,$

We know that $(x-y)^2\equiv (x+y)^2-4xy=26^2-4(144)=676-576=100,x-y=\pm 10\,$

If $x-y=10,x+y=13\,$ by solving we get $x=18,y=8\,$

If $x-y=-10,x+y=16\,$ by solving we get $x=8,y=18\,$

So the numbers are $8,18\,$