1.

difference of the squares of two numbers is 180 the square of the smaller number is 8 times the larger number find the two numbers

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Lets suppose the two numbers are X and Y, where Y is the smaller number of the two, then
x^2 - y^2 = 180-----------1 
the square of the smaller number is eight TIMES the larger number, so
y^2 = 8x--------------------2 
So, using equation 2 in equation 1, we have
x^2 - 8x = 180
x^2 - 8x - 180 = 0
splitting -8x into 10x - 18x
x^2 +10x - 18x - 180 = 0
x(x+10) -18(x+10) = 0
(x - 18)(x +10) = 0 
so x = 18 or x = -10 
If x = 18, using equation 2,
y^2 = 8 X 18 = 144
then y = +-12 
If x = -10, using equation 2,
y^2 = 8 X -10 = -80
then y will be an imaginary number. 
considering you are using only REAL numbers, the solution will be x = 18 and y = +-12. 

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