2x^2+4x=1

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


Simplifying
2x2 + 4x = 1

Reorder the terms:
4x + 2x2 = 1

Solving
4x + 2x2 = 1

Solving for variable 'x'.

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

Combine like terms: 1 + -1 = 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 = 1.5
1 + 2x + x2 = 1.5

Factor a perfect square on the left side:
(x + 1)(x + 1) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(x + 1) equal to 1.224744871 and -1.224744871.

Subproblem 1

x + 1 = 1.224744871 Simplifying x + 1 = 1.224744871 Reorder the terms: 1 + x = 1.224744871 Solving 1 + x = 1.224744871 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 = 1.224744871 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.224744871 + -1 x = 1.224744871 + -1 Combine like terms: 1.224744871 + -1 = 0.224744871 x = 0.224744871 Simplifying x = 0.224744871

Subproblem 2

x + 1 = -1.224744871 Simplifying x + 1 = -1.224744871 Reorder the terms: 1 + x = -1.224744871 Solving 1 + x = -1.224744871 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 = -1.224744871 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.224744871 + -1 x = -1.224744871 + -1 Combine like terms: -1.224744871 + -1 = -2.224744871 x = -2.224744871 Simplifying x = -2.224744871

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.224744871, -2.224744871}

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