<PRE><B>Question 1100680</B>
&lt;br&gt;
The difference between two positive numbers is 223 smaller than the sum of their squares. What are the two numbers?&lt;br&gt;
The equation is
{{{y-x = x^2+y^2-223}}}&lt;br&gt;
The problem states that the numbers are positive; it also SEEMS to imply that the answers are supposed to be integers.  So  let&#39;s see if we can find integer solutions.&lt;br&gt;
{{{y-x = x^2+y^2-223}}}&lt;br&gt;
{{{x^2+x+y^2-y = 223}}}
{{{x^2+x+.25 + y^2-y+.25 = 223.5}}}
{{{(x+.5)^2 + (y-.5)^2 = 223.5}}}&lt;br&gt;
Note this is the equation of a circle, so there ARE positive solutions.  But we are (I think) looking for positive integer solutions.&lt;br&gt;
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.&lt;br&gt;
{{{(x+.5)^2 + (y-.5)^2 = 223.5}}}
{{{(y-.5)^2 = 233.5-(x+.5)^2}}}
{{{y-.5 = sqrt(233.5-(x+.5)^2)}}}
{{{y = sqrt(233.5-(x+.5)^2) + .5}}}&lt;br&gt;
My TI83 calculator shows that there are in fact no integer solutions to the equation.&lt;br&gt;
So my suspicion is that the problem is flawed.&lt;br&gt;
Or, if the problem is stated correctly, then we aren&#39;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.&lt;br&gt;
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.</PRE>