x^2+x=2014

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Solution for x^2+x=2014 equation:


Simplifying
x2 + x = 2014

Reorder the terms:
x + x2 = 2014

Solving
x + x2 = 2014

Solving for variable 'x'.

Reorder the terms:
-2014 + x + x2 = 2014 + -2014

Combine like terms: 2014 + -2014 = 0
-2014 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2014' to each side of the equation.
-2014 + x + 2014 + x2 = 0 + 2014

Reorder the terms:
-2014 + 2014 + x + x2 = 0 + 2014

Combine like terms: -2014 + 2014 = 0
0 + x + x2 = 0 + 2014
x + x2 = 0 + 2014

Combine like terms: 0 + 2014 = 2014
x + x2 = 2014

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 = 2014 + 0.25

Reorder the terms:
0.25 + x + x2 = 2014 + 0.25

Combine like terms: 2014 + 0.25 = 2014.25
0.25 + x + x2 = 2014.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 2014.25

Calculate the square root of the right side: 44.880396611

Break this problem into two subproblems by setting 
(x + 0.5) equal to 44.880396611 and -44.880396611.

Subproblem 1

x + 0.5 = 44.880396611 Simplifying x + 0.5 = 44.880396611 Reorder the terms: 0.5 + x = 44.880396611 Solving 0.5 + x = 44.880396611 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 = 44.880396611 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 44.880396611 + -0.5 x = 44.880396611 + -0.5 Combine like terms: 44.880396611 + -0.5 = 44.380396611 x = 44.380396611 Simplifying x = 44.380396611

Subproblem 2

x + 0.5 = -44.880396611 Simplifying x + 0.5 = -44.880396611 Reorder the terms: 0.5 + x = -44.880396611 Solving 0.5 + x = -44.880396611 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 = -44.880396611 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -44.880396611 + -0.5 x = -44.880396611 + -0.5 Combine like terms: -44.880396611 + -0.5 = -45.380396611 x = -45.380396611 Simplifying x = -45.380396611

Solution

The solution to the problem is based on the solutions from the subproblems. x = {44.380396611, -45.380396611}

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