Answer:
y = sqrt(-2 x - 1/4) + 1/2 (2 i x + i) or y = 1/2 (2 i x + i) - sqrt((-2 x - 1/4))
x = sqrt((-2 i) y + 1/4) + 1/2 (-2 i y + 1) or x = 1/2 (-2 i y + 1) - sqrt(((-2 i) y + 1/4))
Explanation:
Solve for y:
(x + (0 + i) y)^2 = x + (0 - i) y
Subtract -i y + x from both sides:
-x + (i y + x)^2 + i y = 0
Expand and collect in terms of y:
-x + x^2 + y (2 i x + i) - y^2 = 0
Multiply both sides by -1:
x - x^2 + y (-2 i x - i) + y^2 = 0
Subtract -x^2 + x from both sides:
y (-2 i x - i) + y^2 = x^2 - x
Add 1/4 (-2 i x - i)^2 to both sides:
1/4 (-2 i x - i)^2 + y (-2 i x - i) + y^2 = 1/4 (-2 i x - i)^2 - x + x^2
Write the left hand side as a square:
(1/2 (-2 i x - i) + y)^2 = 1/4 (-2 i x - i)^2 - x + x^2
Take the square root of both sides:
1/2 (-2 i x - i) + y = sqrt(1/4 (-2 i x - i)^2 - x + x^2) or 1/2 (-2 i x - i) + y = -sqrt(1/4 (-2 i x - i)^2 - x + x^2)
Subtract 1/2 (-2 i x - i) from both sides:
y = 1/2 (2 i x + i) + sqrt((((-2 i) x - i)^2)/4 - x + x^2) or 1/2 (-2 i x - i) + y = -sqrt(1/4 (-2 i x - i)^2 - x + x^2)
1/4 (-2 i x - i)^2 - x + x^2 = -2 x - 1/4:
y = 1/2 (2 i x + i) + sqrt((-2 x - 1/4)) or 1/2 (-2 i x - i) + y = -sqrt(1/4 (-2 i x - i)^2 - x + x^2)
Subtract 1/2 (-2 i x - i) from both sides:
y = sqrt(-2 x - 1/4) + 1/2 (2 i x + i) or y = 1/2 (2 i x + i) - sqrt((((-2 i) x - i)^2)/4 - x + x^2)
1/4 (-2 i x - i)^2 - x + x^2 = -2 x - 1/4:
Answer: y = sqrt(-2 x - 1/4) + 1/2 (2 i x + i) or y = 1/2 (2 i x + i) - sqrt((-2 x - 1/4))
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Solve for x:
(x + (0 + i) y)^2 = x + (0 - i) y
Subtract -i y + x from both sides:
-x + (i y + x)^2 + i y = 0
Expand and collect in terms of x:
x^2 + x (2 i y - 1) + i y - y^2 = 0
Subtract -y^2 + i y from both sides:
x^2 + x (2 i y - 1) = y^2 - i y
Add 1/4 (2 i y - 1)^2 to both sides:
x^2 + x (2 i y - 1) + 1/4 (2 i y - 1)^2 = 1/4 (2 i y - 1)^2 - i y + y^2
Write the left hand side as a square:
(x + 1/2 (2 i y - 1))^2 = 1/4 (2 i y - 1)^2 - i y + y^2
Take the square root of both sides:
x + 1/2 (2 i y - 1) = sqrt(1/4 (2 i y - 1)^2 - i y + y^2) or x + 1/2 (2 i y - 1) = -sqrt(1/4 (2 i y - 1)^2 - i y + y^2)
Subtract 1/2 (2 i y - 1) from both sides:
x = 1/2 (-2 i y + 1) + sqrt((((2 i) y - 1)^2)/4 - i y + y^2) or x + 1/2 (2 i y - 1) = -sqrt(1/4 (2 i y - 1)^2 - i y + y^2)
1/4 (2 i y - 1)^2 - i y + y^2 = -2 i y + 1/4:
x = 1/2 (-2 i y + 1) + sqrt(((-2 i) y + 1/4)) or x + 1/2 (2 i y - 1) = -sqrt(1/4 (2 i y - 1)^2 - i y + y^2)
Subtract 1/2 (2 i y - 1) from both sides:
x = sqrt((-2 i) y + 1/4) + 1/2 (-2 i y + 1) or x = 1/2 (-2 i y + 1) - sqrt(((2 i y - 1)^2)/4 - i y + y^2)
1/4 (2 i y - 1)^2 - i y + y^2 = -2 i y + 1/4:
Answer: x = sqrt((-2 i) y + 1/4) + 1/2 (-2 i y + 1) or x = 1/2 (-2 i y + 1) - sqrt(((-2 i) y + 1/4))