x(2x+1)=68

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Solution for x(2x+1)=68 equation:


Simplifying
x(2x + 1) = 68

Reorder the terms:
x(1 + 2x) = 68
(1 * x + 2x * x) = 68
(1x + 2x2) = 68

Solving
1x + 2x2 = 68

Solving for variable 'x'.

Reorder the terms:
-68 + 1x + 2x2 = 68 + -68

Combine like terms: 68 + -68 = 0
-68 + 1x + 2x2 = 0

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

Divide each side by '2'.
-34 + 0.5x + x2 = 0

Move the constant term to the right:

Add '34' to each side of the equation.
-34 + 0.5x + 34 + x2 = 0 + 34

Reorder the terms:
-34 + 34 + 0.5x + x2 = 0 + 34

Combine like terms: -34 + 34 = 0
0 + 0.5x + x2 = 0 + 34
0.5x + x2 = 0 + 34

Combine like terms: 0 + 34 = 34
0.5x + x2 = 34

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 = 34 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 34 + 0.0625

Combine like terms: 34 + 0.0625 = 34.0625
0.0625 + 0.5x + x2 = 34.0625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 34.0625

Calculate the square root of the right side: 5.836308765

Break this problem into two subproblems by setting 
(x + 0.25) equal to 5.836308765 and -5.836308765.

Subproblem 1

x + 0.25 = 5.836308765 Simplifying x + 0.25 = 5.836308765 Reorder the terms: 0.25 + x = 5.836308765 Solving 0.25 + x = 5.836308765 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 = 5.836308765 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.836308765 + -0.25 x = 5.836308765 + -0.25 Combine like terms: 5.836308765 + -0.25 = 5.586308765 x = 5.586308765 Simplifying x = 5.586308765

Subproblem 2

x + 0.25 = -5.836308765 Simplifying x + 0.25 = -5.836308765 Reorder the terms: 0.25 + x = -5.836308765 Solving 0.25 + x = -5.836308765 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 = -5.836308765 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.836308765 + -0.25 x = -5.836308765 + -0.25 Combine like terms: -5.836308765 + -0.25 = -6.086308765 x = -6.086308765 Simplifying x = -6.086308765

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.586308765, -6.086308765}

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