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

Question 201238: The sum of two squares of two numbers is 128, the product of the two numbers is 64. what are the 2 numbers?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
"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

--------------------------

Now let's find "y" when :

Go back to the first isolated equation.

Plug in

Reduce.

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

--------------------------

Now let's find "y" when :

Go back to the first isolated equation.

Plug in

Reduce.

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

===============================================================

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
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