SOLUTION: The difference between two positive numbers is 223 smaller than the sum of their squares. What are the two numbers?

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Question 1100680: The difference between two positive numbers is 223 smaller than the sum of their squares. What are the two numbers?
Found 3 solutions by richwmiller, ikleyn, greenestamps:
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
x-y=x^2+y^2-223
It is obvious to anyone that this equation can't be solved with values for both x and y because there are two unknowns and one equation.
Why would a tutor even think that it needs to be mentioned?!!

Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
.
The condition gives ONLY ONE EQUATION for two unknown real numbers.

It is not sufficient to get a unique answer.



Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


The difference between two positive numbers is 223 smaller than the sum of their squares. What are the two numbers?

The equation is
y-x+=+x%5E2%2By%5E2-223

The problem states that the numbers are positive; it also SEEMS to imply that the answers are supposed to be integers. So let's see if we can find integer solutions.

y-x+=+x%5E2%2By%5E2-223

x%5E2%2Bx%2By%5E2-y+=+223
x%5E2%2Bx%2B.25+%2B+y%5E2-y%2B.25+=+223.5
%28x%2B.5%29%5E2+%2B+%28y-.5%29%5E2+=+223.5

Note this is the equation of a circle, so there ARE positive solutions. But we are (I think) looking for positive integer solutions.

To search for solutions, we can solve the equation for one variable in terms of the other; then we can use a software tool such as the table feature of a graphing calculator, or perhaps a spreadsheet, to look for integer solutions.

%28x%2B.5%29%5E2+%2B+%28y-.5%29%5E2+=+223.5
%28y-.5%29%5E2+=+233.5-%28x%2B.5%29%5E2
y-.5+=+sqrt%28233.5-%28x%2B.5%29%5E2%29
y+=+sqrt%28233.5-%28x%2B.5%29%5E2%29+%2B+.5

My TI83 calculator shows that there are in fact no integer solutions to the equation.

So my suspicion is that the problem is flawed.

Or, if the problem is stated correctly, then we aren't looking for integer solutions. And in that case, we can find an infinite number of solutions, remembering that the equation is the equation of a circle.

That circle has center at (-.5,.5) and a radius of just under 15. So we can choose any value of x or y up to about 14 or 15 and solve for the corresponding value of the other variable.