2x(x+1)=1

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Solution for 2x(x+1)=1 equation:


Simplifying
2x(x + 1) = 1

Reorder the terms:
2x(1 + x) = 1
(1 * 2x + x * 2x) = 1
(2x + 2x2) = 1

Solving
2x + 2x2 = 1

Solving for variable 'x'.

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

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

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

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

Move the constant term to the right:

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

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

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

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

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.
 + 0.25 + x2 = 0.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 0.5 + 0.25

Combine like terms: 0.5 + 0.25 = 0.75
1.25 + x2 = 0.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

x + 0.5 = 0.866025404 Simplifying x + 0.5 = 0.866025404 Reorder the terms: 0.5 + x = 0.866025404 Solving 0.5 + x = 0.866025404 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 = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.866025404 + -0.5 x = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 x = 0.366025404 Simplifying x = 0.366025404

Subproblem 2

x + 0.5 = -0.866025404 Simplifying x + 0.5 = -0.866025404 Reorder the terms: 0.5 + x = -0.866025404 Solving 0.5 + x = -0.866025404 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 = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.866025404 + -0.5 x = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 x = -1.366025404 Simplifying x = -1.366025404

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.366025404, -1.366025404}

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