Question 896475
the solution is, i believe, a = 11 and b = 6


you get (a+b)*(a-b) = (11+6) * (11-6) = 17 * 5 = 85


i solved it graphically.


i couldn't see any other way except to create a table and then populate the table until you got a number for a that was a whole number.


there are lots of values, but only 2 values that i could find that were whole numbers.


those values were a = 11 and b = 6


the other values were a = 43 and b = 42


since the difference could not be equal to 1, the only other number i could find was a = 11 and b = 6.


since (a+b)*(a-b) = a^2 - b^2, i solved for a^2 to get a^2 = b^2 + 85


what was left then was to find a number of b^2 + 85 that was a perfect square.


perfect squares greater than 85 were


100
121
144
etc


subtracting 85 from them needs to yield a perfect square as well.


100 - 85 = 15 which is not a perfect square.
121 - 85 = 36 which is a perfect square.
144 - 85 = 59 which is not a perfect square.


36 was the one.
that led to b = 6


that also led to a = 11 because sqrt(121) = 11 and sqrt(36+85) = 11


i then got a^2 - b^2 = 85 becoming 121 - 36 = 85 becoming 85 = 85 confirming that the solution was correct.


graphing was the cleanest and easiest way to find the answer, but you needed to be able to see that the solution was a hole number.


creating an automated table helped with that.
it clearly showed that when x = 6, y = 11


the equation that was graphed was y = sqrt(x^2 + 85)