The lcm of 2 numbers is 60 and 1 of the numbers is 7 less than the other number. what are the numbers?

Let's represent the two numbers by x and y.  Then xy=60.  The smaller number here is x=y-7.

Then (y-7)y=60, or y^2 - 7y - 60 = 0.  Use the quadratic formula to (1) determine whether y has real values and (2) to determine those values if they are real:

discriminant = b^2 - 4ac; here the discriminant is (-7)^2 - 4(1)(-60) = 191.  Because the discriminant is positive, this equation has two real, unequal roots, which are
        -(-7) + sqrt(191)
y =
             -2(1)

and 

        -(-7) - sqrt(191)
y = = 3.41 (approximately)
             -2(1)

Unfortunately, this doesn't make sense, since the LCM of two numbers is generally an integer.

Try thinking this way:  If the LCM is 60, then xy = 60.  What would happen if x=5 and y=12?  Is xy = 60?  Yes.  Is 5 seven less than 12?  Yes.