x+2*x^2-416=0

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Solution for x+2*x^2-416=0 equation:


Simplifying
x + 2x2 + -416 = 0

Reorder the terms:
-416 + x + 2x2 = 0

Solving
-416 + x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-208 + 0.5x + x2 = 0

Move the constant term to the right:

Add '208' to each side of the equation.
-208 + 0.5x + 208 + x2 = 0 + 208

Reorder the terms:
-208 + 208 + 0.5x + x2 = 0 + 208

Combine like terms: -208 + 208 = 0
0 + 0.5x + x2 = 0 + 208
0.5x + x2 = 0 + 208

Combine like terms: 0 + 208 = 208
0.5x + x2 = 208

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 = 208 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 208 + 0.25

Combine like terms: 208 + 0.25 = 208.25
0.25 + 0.5x + x2 = 208.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 208.25

Calculate the square root of the right side: 14.43086969

Break this problem into two subproblems by setting 
(x + 0.5) equal to 14.43086969 and -14.43086969.

Subproblem 1

x + 0.5 = 14.43086969 Simplifying x + 0.5 = 14.43086969 Reorder the terms: 0.5 + x = 14.43086969 Solving 0.5 + x = 14.43086969 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 = 14.43086969 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 14.43086969 + -0.5 x = 14.43086969 + -0.5 Combine like terms: 14.43086969 + -0.5 = 13.93086969 x = 13.93086969 Simplifying x = 13.93086969

Subproblem 2

x + 0.5 = -14.43086969 Simplifying x + 0.5 = -14.43086969 Reorder the terms: 0.5 + x = -14.43086969 Solving 0.5 + x = -14.43086969 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 = -14.43086969 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -14.43086969 + -0.5 x = -14.43086969 + -0.5 Combine like terms: -14.43086969 + -0.5 = -14.93086969 x = -14.93086969 Simplifying x = -14.93086969

Solution

The solution to the problem is based on the solutions from the subproblems. x = {13.93086969, -14.93086969}

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