Simplify : sq root of{(x+3)^2+y^2} + sq root of {(X-3)^2 + Y^2}= 8

1) √{(x+3)² + y²} + √{(x-3)² + y²} = 8 [Given]

2) ==> √{(x+3)² + y²} = 8 - √{(x-3)² + y²}

3) Squaring both sides, {(x+3)² + y²} = 64 + {(x-3)² + y²} - 16√{(x-3)² + y²}

4) Expanding and simplifying, 3x - 16 = -4√{(x-3)² + y²}

5) Again squaring both sides, 9x² - 96x + 256 = 16(x² - 6x + 9 + y²)

6) Expanding and simplifying, 7x² + 16y² = 112

The above is the ellipse equation of the form, (x²/16) + (y²/7) = 1, whose center is (0,0) and Major axis = 8 and Minor axis = 2√7.

square both sides:

(x + 3)^2 + y^2 + (x - 3)^2 + y^2 + 2 * sqrt( ((x + 3)^2 + y^2) * ((x - 3)^2 + y^2) ) = 64
x^2 + 6x + 9 + x^2 - 6x + 9 + 2y^2 + 2 * sqrt( ((x + 3)^2 + y^2) * ((x - 3)^2 + y^2) ) = 64
2x^2 + 18 + 2y^2 + 2 * sqrt( ((x + 3)^2 + y^2) * ((x - 3)^2 + y^2) ) = 64
x^2 + y^2 + 9 + sqrt( ((x + 3)^2 + y^2) * ((x -3)^2 + y^2) ) = 32
x^2 + y^2 - 23 = -sqrt( ((x + 3)^2 + y^2) * ((x - 3)^2 + y^2) )

Square both sides again:

(x^2 + y^2 - 23)^2 = ((x + 3)^2 + y^2) * ((x - 3)^2 + y^2))
(x^2 + y^2 - 23)^2 = (x + 3)^2 * (x - 3)^2 + (x + 3)^2 * y^2 + y^2 * (x - 3)^2 + y^4
(x^2 + y^2 - 23)^2 = (x^2 - 9)^2 + y^2 * ((x - 3)^2 + (x + 3)^2) + y^4
(x^2 + y^2 - 23)^2 = (x^2 - 9)^2 + y^2 * (x^2 - 6x + 9 + x^2 + 6x + 9) + y^4
(x^2 + y^2 - 23)^2 = (x^4 - 18x^2 + 81) + y^2 * (2x^2 + 18) + y^4
x^4 + 2x^2 * y^2 - 46x^2 + y^4 - 46y^2 + 529 = x^4 + y^4 - 18x^2 + 81 + 2x^2 * y^2 + 18y^2
x^4 - x^4 + y^4 - y^4 + 2x^2 * y^2 - 2x^2 * y^2 - 46x^2 - 46y^2 - 18y^2 + 529 - 81 = 0
-46x^2 - 64y^2 + 448 = 0
23x^2 + 32y^2 - 224 = 0
32y^2 = 224 - 23x^2
y^2 = (224 - 23x^2) / 32
y = +/- sqrt( (224 - 23x^2) / 32 )

I got that it simplifies down to x + y = 4.