SOLUTION: the sum of two positive integers is 30 and their product is 216, find the positive difference between the two numbers

Question 573587: the sum of two positive integers is 30 and their product is 216, find the positive difference between the two numbers
Answer by mathsmiles(68) (Show Source): You can put this solution on YOUR website!
Let's let X and Y be the two numbers we need to search for.
Their sum is 30 so
X + Y = 30
their product is 216 so
XY = 216

Let's use substitution method, so need to get X or Y by itself in one of these equations. I'll pick the 1st equation since you asked for my help and I get some perk for doing this. :-)

X + Y = 30
Subtracting Y from both sides:
X = 30 - Y

Now substitute this into the 2nd equation for X:
XY = 216
(30-Y)(Y) = 216
Multiplying out:
30Y - Y^2 = 216 Let's move everything to the right
-30Y + Y^2 -30Y + Y^2
---------------------------
0 = Y^2 - 30Y + 216

(Y - )(Y - ) Need to find two factors of 216 which add up to 30. (Ugh! This is the worst part!!)
The factors of 216 are:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
Lucky for me, I found
http://www.calculatorsoup.com/calculators/math/factors.php

See two factors which add to 30 and also multiply to get 216?
18 and 12

(Y - 18)(Y - 12) = 0
Let's check and make sure we get the original equation using FOIL:
Y^2 - 12Y - 18Y + 216 = Y^2 - 30Y + 216 Ok, good.

Solving for Y:
Y - 18 = 0
Y = 18
AND
Y - 12 = 0
Y = 12
Looks like we have our two numbers. Now go back to the original question:

Find the positive difference between the two numbers:
18 and 12
18 - 12 = 6 That's the answer!

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