x^2+x-365=0

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Solution for x^2+x-365=0 equation:


Simplifying
x2 + x + -365 = 0

Reorder the terms:
-365 + x + x2 = 0

Solving
-365 + x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '365' to each side of the equation.
-365 + x + 365 + x2 = 0 + 365

Reorder the terms:
-365 + 365 + x + x2 = 0 + 365

Combine like terms: -365 + 365 = 0
0 + x + x2 = 0 + 365
x + x2 = 0 + 365

Combine like terms: 0 + 365 = 365
x + x2 = 365

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 = 365 + 0.25

Reorder the terms:
0.25 + x + x2 = 365 + 0.25

Combine like terms: 365 + 0.25 = 365.25
0.25 + x + x2 = 365.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 365.25

Calculate the square root of the right side: 19.111514854

Break this problem into two subproblems by setting 
(x + 0.5) equal to 19.111514854 and -19.111514854.

Subproblem 1

x + 0.5 = 19.111514854 Simplifying x + 0.5 = 19.111514854 Reorder the terms: 0.5 + x = 19.111514854 Solving 0.5 + x = 19.111514854 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 = 19.111514854 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 19.111514854 + -0.5 x = 19.111514854 + -0.5 Combine like terms: 19.111514854 + -0.5 = 18.611514854 x = 18.611514854 Simplifying x = 18.611514854

Subproblem 2

x + 0.5 = -19.111514854 Simplifying x + 0.5 = -19.111514854 Reorder the terms: 0.5 + x = -19.111514854 Solving 0.5 + x = -19.111514854 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 = -19.111514854 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -19.111514854 + -0.5 x = -19.111514854 + -0.5 Combine like terms: -19.111514854 + -0.5 = -19.611514854 x = -19.611514854 Simplifying x = -19.611514854

Solution

The solution to the problem is based on the solutions from the subproblems. x = {18.611514854, -19.611514854}

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