Question 1209119
<font color=black size=3>
Answer: <font color=red>11061</font>


Explanation


Let's say x and y are the two numbers where x is the bigger value.
x > y and both are larger than 0
<font color=blue>xy = 630</font> is the product of the values which will be useful later.


x-y = 99 since the two numbers are 99 units apart, ie. their difference is 99


Since your teacher wants to know the sum of the squares, let's square both sides to see what happens.
(x-y)^2 = 99^2
x^2-2xy+y^2 = 9801
x^2+y^2-2xy = 9801
x^2+y^2-2*(<font color=blue>xy</font>) = 9801
x^2+y^2-2*(<font color=blue>630</font>) = 9801 ..... plug in <font color=blue>xy = 630</font>
x^2+y^2-1260 = 9801
x^2+y^2 = 9801+1260
x^2+y^2 = <font color=red>11061</font> is the sum of the squares of the two mystery numbers. 


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Notice how we don't need to find x and y themselves. 
However, if you wanted to, you could follow that pathway. 
Ignore this section if you prefer the previous method shown above.


x-y = 99 can be rearranged to y = x-99
Plug that into the other equation to get,
xy = 630
x*( y ) = 630
x*(x-99) = 630
x^2-99x = 630
x^2-99x-630 = 0


Then apply the quadratic formula to get the roots x = -6 and x = 105. 
I'll let the student handle the scratch work.
We ignore x = -6 because the instructions mention the numbers are positive.


If x = 105 then y = x-99 = 105-99 = 6.
We determine that x = 105 and y = 6 are the two mystery numbers.
The scratch work shown below confirms we have the correct x,y pairing.
<pre>x-y = 105-6 = 99
x*y = 105*6 = 630</pre>Then,
x^2+y^2 = 105^2+6^2 = <font color=red>11061</font> is the final answer.
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