2x(2x+1)=1

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


Simplifying
2x(2x + 1) = 1

Reorder the terms:
2x(1 + 2x) = 1
(1 * 2x + 2x * 2x) = 1
(2x + 4x2) = 1

Solving
2x + 4x2 = 1

Solving for variable 'x'.

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

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

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

Divide each side by '4'.
-0.25 + 0.5x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 0.5x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 0.5x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 0.5x + x2 = 0 + 0.25
0.5x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
0.5x + x2 = 0.25

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 0.25 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 0.25 + 0.0625

Combine like terms: 0.25 + 0.0625 = 0.3125
0.0625 + 0.5x + x2 = 0.3125

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(x + 0.25) equal to 0.559016994 and -0.559016994.

Subproblem 1

x + 0.25 = 0.559016994 Simplifying x + 0.25 = 0.559016994 Reorder the terms: 0.25 + x = 0.559016994 Solving 0.25 + x = 0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.559016994 + -0.25 x = 0.559016994 + -0.25 Combine like terms: 0.559016994 + -0.25 = 0.309016994 x = 0.309016994 Simplifying x = 0.309016994

Subproblem 2

x + 0.25 = -0.559016994 Simplifying x + 0.25 = -0.559016994 Reorder the terms: 0.25 + x = -0.559016994 Solving 0.25 + x = -0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.559016994 + -0.25 x = -0.559016994 + -0.25 Combine like terms: -0.559016994 + -0.25 = -0.809016994 x = -0.809016994 Simplifying x = -0.809016994

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.309016994, -0.809016994}

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