2x^2+1=-4x

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Solution for 2x^2+1=-4x equation:


Simplifying
2x2 + 1 = -4x

Reorder the terms:
1 + 2x2 = -4x

Solving
1 + 2x2 = -4x

Solving for variable 'x'.

Reorder the terms:
1 + 4x + 2x2 = -4x + 4x

Combine like terms: -4x + 4x = 0
1 + 4x + 2x2 = 0

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

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

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2x + x2 = 0 + -0.5
2x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2x + x2 = -0.5

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = -0.5 + 1

Reorder the terms:
1 + 2x + x2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + 2x + x2 = 0.5

Factor a perfect square on the left side:
(x + 1)(x + 1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + 1) equal to 0.707106781 and -0.707106781.

Subproblem 1

x + 1 = 0.707106781 Simplifying x + 1 = 0.707106781 Reorder the terms: 1 + x = 0.707106781 Solving 1 + x = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 0.707106781 + -1 Combine like terms: 1 + -1 = 0 0 + x = 0.707106781 + -1 x = 0.707106781 + -1 Combine like terms: 0.707106781 + -1 = -0.292893219 x = -0.292893219 Simplifying x = -0.292893219

Subproblem 2

x + 1 = -0.707106781 Simplifying x + 1 = -0.707106781 Reorder the terms: 1 + x = -0.707106781 Solving 1 + x = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -0.707106781 + -1 Combine like terms: 1 + -1 = 0 0 + x = -0.707106781 + -1 x = -0.707106781 + -1 Combine like terms: -0.707106781 + -1 = -1.707106781 x = -1.707106781 Simplifying x = -1.707106781

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.292893219, -1.707106781}

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