0=x^2-x-405

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Solution for 0=x^2-x-405 equation:


Simplifying
0 = x2 + -1x + -405

Reorder the terms:
0 = -405 + -1x + x2

Solving
0 = -405 + -1x + x2

Solving for variable 'x'.

Combine like terms: 0 + 405 = 405
405 + x + -1x2 = -405 + -1x + x2 + 405 + x + -1x2

Reorder the terms:
405 + x + -1x2 = -405 + 405 + -1x + x + x2 + -1x2

Combine like terms: -405 + 405 = 0
405 + x + -1x2 = 0 + -1x + x + x2 + -1x2
405 + x + -1x2 = -1x + x + x2 + -1x2

Combine like terms: -1x + x = 0
405 + x + -1x2 = 0 + x2 + -1x2
405 + x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
405 + x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-405 + -1x + x2 = 0

Move the constant term to the right:

Add '405' to each side of the equation.
-405 + -1x + 405 + x2 = 0 + 405

Reorder the terms:
-405 + 405 + -1x + x2 = 0 + 405

Combine like terms: -405 + 405 = 0
0 + -1x + x2 = 0 + 405
-1x + x2 = 0 + 405

Combine like terms: 0 + 405 = 405
-1x + x2 = 405

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.
-1x + 0.25 + x2 = 405 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 405 + 0.25

Combine like terms: 405 + 0.25 = 405.25
0.25 + -1x + x2 = 405.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 405.25

Calculate the square root of the right side: 20.130822139

Break this problem into two subproblems by setting 
(x + 0.5) equal to 20.130822139 and -20.130822139.

Subproblem 1

x + 0.5 = 20.130822139 Simplifying x + 0.5 = 20.130822139 Reorder the terms: 0.5 + x = 20.130822139 Solving 0.5 + x = 20.130822139 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 = 20.130822139 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 20.130822139 + -0.5 x = 20.130822139 + -0.5 Combine like terms: 20.130822139 + -0.5 = 19.630822139 x = 19.630822139 Simplifying x = 19.630822139

Subproblem 2

x + 0.5 = -20.130822139 Simplifying x + 0.5 = -20.130822139 Reorder the terms: 0.5 + x = -20.130822139 Solving 0.5 + x = -20.130822139 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 = -20.130822139 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -20.130822139 + -0.5 x = -20.130822139 + -0.5 Combine like terms: -20.130822139 + -0.5 = -20.630822139 x = -20.630822139 Simplifying x = -20.630822139

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.630822139, -20.630822139}

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