x^2+x=256

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Solution for x^2+x=256 equation:


Simplifying
x2 + x = 256

Reorder the terms:
x + x2 = 256

Solving
x + x2 = 256

Solving for variable 'x'.

Reorder the terms:
-256 + x + x2 = 256 + -256

Combine like terms: 256 + -256 = 0
-256 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '256' to each side of the equation.
-256 + x + 256 + x2 = 0 + 256

Reorder the terms:
-256 + 256 + x + x2 = 0 + 256

Combine like terms: -256 + 256 = 0
0 + x + x2 = 0 + 256
x + x2 = 0 + 256

Combine like terms: 0 + 256 = 256
x + x2 = 256

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 256 + 0.25

Reorder the terms:
0.25 + x + x2 = 256 + 0.25

Combine like terms: 256 + 0.25 = 256.25
0.25 + x + x2 = 256.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 256.25

Calculate the square root of the right side: 16.007810594

Break this problem into two subproblems by setting 
(x + 0.5) equal to 16.007810594 and -16.007810594.

Subproblem 1

x + 0.5 = 16.007810594 Simplifying x + 0.5 = 16.007810594 Reorder the terms: 0.5 + x = 16.007810594 Solving 0.5 + x = 16.007810594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 16.007810594 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 16.007810594 + -0.5 x = 16.007810594 + -0.5 Combine like terms: 16.007810594 + -0.5 = 15.507810594 x = 15.507810594 Simplifying x = 15.507810594

Subproblem 2

x + 0.5 = -16.007810594 Simplifying x + 0.5 = -16.007810594 Reorder the terms: 0.5 + x = -16.007810594 Solving 0.5 + x = -16.007810594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -16.007810594 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -16.007810594 + -0.5 x = -16.007810594 + -0.5 Combine like terms: -16.007810594 + -0.5 = -16.507810594 x = -16.507810594 Simplifying x = -16.507810594

Solution

The solution to the problem is based on the solutions from the subproblems. x = {15.507810594, -16.507810594}

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