x^2+x=412

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Solution for x^2+x=412 equation:


Simplifying
x2 + x = 412

Reorder the terms:
x + x2 = 412

Solving
x + x2 = 412

Solving for variable 'x'.

Reorder the terms:
-412 + x + x2 = 412 + -412

Combine like terms: 412 + -412 = 0
-412 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '412' to each side of the equation.
-412 + x + 412 + x2 = 0 + 412

Reorder the terms:
-412 + 412 + x + x2 = 0 + 412

Combine like terms: -412 + 412 = 0
0 + x + x2 = 0 + 412
x + x2 = 0 + 412

Combine like terms: 0 + 412 = 412
x + x2 = 412

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 = 412 + 0.25

Reorder the terms:
0.25 + x + x2 = 412 + 0.25

Combine like terms: 412 + 0.25 = 412.25
0.25 + x + x2 = 412.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 412.25

Calculate the square root of the right side: 20.303940504

Break this problem into two subproblems by setting 
(x + 0.5) equal to 20.303940504 and -20.303940504.

Subproblem 1

x + 0.5 = 20.303940504 Simplifying x + 0.5 = 20.303940504 Reorder the terms: 0.5 + x = 20.303940504 Solving 0.5 + x = 20.303940504 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 = 20.303940504 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 20.303940504 + -0.5 x = 20.303940504 + -0.5 Combine like terms: 20.303940504 + -0.5 = 19.803940504 x = 19.803940504 Simplifying x = 19.803940504

Subproblem 2

x + 0.5 = -20.303940504 Simplifying x + 0.5 = -20.303940504 Reorder the terms: 0.5 + x = -20.303940504 Solving 0.5 + x = -20.303940504 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 = -20.303940504 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -20.303940504 + -0.5 x = -20.303940504 + -0.5 Combine like terms: -20.303940504 + -0.5 = -20.803940504 x = -20.803940504 Simplifying x = -20.803940504

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.803940504, -20.803940504}

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