x(2x+1)=92

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Solution for x(2x+1)=92 equation:


Simplifying
x(2x + 1) = 92

Reorder the terms:
x(1 + 2x) = 92
(1 * x + 2x * x) = 92
(1x + 2x2) = 92

Solving
1x + 2x2 = 92

Solving for variable 'x'.

Reorder the terms:
-92 + 1x + 2x2 = 92 + -92

Combine like terms: 92 + -92 = 0
-92 + 1x + 2x2 = 0

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

Divide each side by '2'.
-46 + 0.5x + x2 = 0

Move the constant term to the right:

Add '46' to each side of the equation.
-46 + 0.5x + 46 + x2 = 0 + 46

Reorder the terms:
-46 + 46 + 0.5x + x2 = 0 + 46

Combine like terms: -46 + 46 = 0
0 + 0.5x + x2 = 0 + 46
0.5x + x2 = 0 + 46

Combine like terms: 0 + 46 = 46
0.5x + x2 = 46

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 = 46 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 46 + 0.0625

Combine like terms: 46 + 0.0625 = 46.0625
0.0625 + 0.5x + x2 = 46.0625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 46.0625

Calculate the square root of the right side: 6.78693598

Break this problem into two subproblems by setting 
(x + 0.25) equal to 6.78693598 and -6.78693598.

Subproblem 1

x + 0.25 = 6.78693598 Simplifying x + 0.25 = 6.78693598 Reorder the terms: 0.25 + x = 6.78693598 Solving 0.25 + x = 6.78693598 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 = 6.78693598 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 6.78693598 + -0.25 x = 6.78693598 + -0.25 Combine like terms: 6.78693598 + -0.25 = 6.53693598 x = 6.53693598 Simplifying x = 6.53693598

Subproblem 2

x + 0.25 = -6.78693598 Simplifying x + 0.25 = -6.78693598 Reorder the terms: 0.25 + x = -6.78693598 Solving 0.25 + x = -6.78693598 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 = -6.78693598 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -6.78693598 + -0.25 x = -6.78693598 + -0.25 Combine like terms: -6.78693598 + -0.25 = -7.03693598 x = -7.03693598 Simplifying x = -7.03693598

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.53693598, -7.03693598}

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