f(x+y)=f(x)+f(y)+xy+1

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Solution for f(x+y)=f(x)+f(y)+xy+1 equation:


Simplifying
f(x + y) = f(x) + f(y) + xy + 1
(x * f + y * f) = f(x) + f(y) + xy + 1
(fx + fy) = f(x) + f(y) + xy + 1

Multiply f * x
fx + fy = fx + f(y) + xy + 1

Multiply f * y
fx + fy = fx + fy + xy + 1

Reorder the terms:
fx + fy = 1 + fx + fy + xy

Add '-1fx' to each side of the equation.
fx + -1fx + fy = 1 + fx + fy + -1fx + xy

Combine like terms: fx + -1fx = 0
0 + fy = 1 + fx + fy + -1fx + xy
fy = 1 + fx + fy + -1fx + xy

Reorder the terms:
fy = 1 + fx + -1fx + fy + xy

Combine like terms: fx + -1fx = 0
fy = 1 + 0 + fy + xy
fy = 1 + fy + xy

Add '-1fy' to each side of the equation.
fy + -1fy = 1 + fy + -1fy + xy

Combine like terms: fy + -1fy = 0
0 = 1 + fy + -1fy + xy

Combine like terms: fy + -1fy = 0
0 = 1 + 0 + xy
0 = 1 + xy

Solving
0 = 1 + xy

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1xy' to each side of the equation.
0 + -1xy = 1 + xy + -1xy
Remove the zero:
-1xy = 1 + xy + -1xy

Combine like terms: xy + -1xy = 0
-1xy = 1 + 0
-1xy = 1

Divide each side by '-1y'.
x = -1y-1

Simplifying
x = -1y-1

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