The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Explanation

Let the required numbers be x and y such that `xgty`.
Then, `x^(2)-y^(2)=180." "…(i)`
And, `y^(2)=8x." "…(ii)`
From (i) and (ii), we get
`x^(2)-8x-180=0`
`implies" "x^(2)-18x+10x-180=0impliesx(x-18)+10(x-18)=0`
`implies" "(x-18)(x+10)=0impliesx-18=0" or "x+10=0`
`implies" "x=18" or "x=-10.`
Now, `x=18impliesy^(2)=(8xx18)=144`
`implies" "y=12" or "y=-12.`
Also, `x=-10impliesy^(2)={8xx(-10)}=-80,` which is not possible.
Hence, the numbers are `(18" and "12)` or `(18" and "-12).`