SOLUTION: The sum of two squares of two numbers is 128, the product of the two numbers is 64. what are the 2 numbers?

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"The sum of the squares of two numbers is 128" translates to


"The product of the numbers is 64" translates to


Start with the second equation.


Divide both sides by "x".


Move onto the first equation


Plug in


Square to get


Multiply EVERY term by the LCD to clear out the fractions.


Subtract from both sides.


Rearrange the terms.


Let . So


Replace with . Replace with


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "z":


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the only solution (in terms of "z") is


Now recall that we let . So this means that and that or

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Now let's find "y" when :


Go back to the first isolated equation.


Plug in


Reduce.


So when , giving the ordered pair (8,8)



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Now let's find "y" when :


Go back to the first isolated equation.


Plug in


Reduce.


So when , giving the ordered pair (-8,-8)


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Answer:

So the two ordered pair solutions are (8,8) and (-8,-8)



This means that the two numbers are:

8 and 8,

OR

-8 and -8