2x^2+x=405

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Solution for 2x^2+x=405 equation:


Simplifying
2x2 + x = 405

Reorder the terms:
x + 2x2 = 405

Solving
x + 2x2 = 405

Solving for variable 'x'.

Reorder the terms:
-405 + x + 2x2 = 405 + -405

Combine like terms: 405 + -405 = 0
-405 + x + 2x2 = 0

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

Divide each side by '2'.
-202.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '202.5' to each side of the equation.
-202.5 + 0.5x + 202.5 + x2 = 0 + 202.5

Reorder the terms:
-202.5 + 202.5 + 0.5x + x2 = 0 + 202.5

Combine like terms: -202.5 + 202.5 = 0.0
0.0 + 0.5x + x2 = 0 + 202.5
0.5x + x2 = 0 + 202.5

Combine like terms: 0 + 202.5 = 202.5
0.5x + x2 = 202.5

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.
0.5x + 0.25 + x2 = 202.5 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 202.5 + 0.25

Combine like terms: 202.5 + 0.25 = 202.75
0.25 + 0.5x + x2 = 202.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 202.75

Calculate the square root of the right side: 14.239030866

Break this problem into two subproblems by setting 
(x + 0.5) equal to 14.239030866 and -14.239030866.

Subproblem 1

x + 0.5 = 14.239030866 Simplifying x + 0.5 = 14.239030866 Reorder the terms: 0.5 + x = 14.239030866 Solving 0.5 + x = 14.239030866 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 = 14.239030866 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 14.239030866 + -0.5 x = 14.239030866 + -0.5 Combine like terms: 14.239030866 + -0.5 = 13.739030866 x = 13.739030866 Simplifying x = 13.739030866

Subproblem 2

x + 0.5 = -14.239030866 Simplifying x + 0.5 = -14.239030866 Reorder the terms: 0.5 + x = -14.239030866 Solving 0.5 + x = -14.239030866 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 = -14.239030866 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -14.239030866 + -0.5 x = -14.239030866 + -0.5 Combine like terms: -14.239030866 + -0.5 = -14.739030866 x = -14.739030866 Simplifying x = -14.739030866

Solution

The solution to the problem is based on the solutions from the subproblems. x = {13.739030866, -14.739030866}

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