SOLUTION: Let x and y be complex numbers. If x + y = 2 and xy = 7 - x^2 - y^2, then what is x^2 + y^2?

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Question 1209394: Let x and y be complex numbers. If x + y = 2 and xy = 7 - x^2 - y^2, then what is x^2 + y^2?
Found 3 solutions by ikleyn, math_tutor2020, Edwin McCravy:
Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
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Let x and y be complex numbers. If x + y = 2 and xy = 7 - x^2 - y^2, then what is x^2 + y^2?
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x^2 + y^2 = add and subtract 2xy = (x^2 + 2xy + y^2) - 2xy = first parentheses is the square = (x+y)^2 - 2xy = 


          = substitute the given values = 2^2 - 2*(7-x^2-y^2).


Thus you have now

    x^2 + y^2 = 4 - 14 + 2(x^2 + y^2),

    x^2 + y^2 = -10 + 2(x^2 + y^2),

     10 = x^2 + y^2.


At this point, the problem is solved completely.


ANSWER.  x^2 + y^2 = 10.

Solved.



Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Answer: 10


Work Shown
xy = 7 - x^2 - y^2
xy = 7 - (x^2 + y^2)
x^2 + y^2 = 7 - xy
2(x^2 + y^2) = 14 - 2xy
Let's call this equation (3)

x + y = 2
(x + y)^2 = 2^2
x^2 + 2xy + y^2 = 4
x^2 + y^2 = 4 - 2xy
Let's call this equation (4)

Subtract straight down for equations (3) and (4) to arrive at
x^2 + y^2 = 10

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An alternative approach would be to rearrange x+y = 2 into y = 2-x
Then,
xy = 7-x^2-y^2
x(2-x) = 7-x^2-(2-x)^2
Solving that for x yields x = -1 and x = 3
I'll let the student handle the scratch work.

If x = -1 then y = 2-x = 2-(-1) = 3
If x = 3 then y = 2-x = 2-3 = -1
We get the same pair of values just in a different order.
Turns out the order doesn't matter.

Then,
x^2 + y^2 = (-1)^2 + (3)^2 = 10
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
As the last tutor showed, 

{x,y} = {3,-1} and x2 + y2 = 9 + 1 = 10

What the student should observe here is that 

All REAL numbers ARE COMPLEX numbers!

3 and -1 are complex numbers because they are 3 + 0i and -1 + 0i respectively.

Nobody said the coefficient of " i " cannot be 0. And if it's 0, the number
is still complex.  It's both REAL and COMPLEX.

Edwin
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