If the square of a positive number is increased by seven times the number, the result is eight. Find the number.

1^2 + 7*1 =8 

...so 1 will work.

In general:
x^2+7x=8 leads to
x^2+7x-8=0
which factors to:
(x-1)(x+8)=0
but -8 is not positive