(x+1)[2x+1]=x+0.5

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Solution for (x+1)[2x+1]=x+0.5 equation:


Simplifying
(x + 1)[2x + 1] = x + 0.5

Reorder the terms:
(1 + x)[2x + 1] = x + 0.5

Reorder the terms:
(1 + x)[1 + 2x] = x + 0.5

Multiply (1 + x) * [1 + 2x]
(1[1 + 2x] + x[1 + 2x]) = x + 0.5
([1 * 1 + 2x * 1] + x[1 + 2x]) = x + 0.5
([1 + 2x] + x[1 + 2x]) = x + 0.5
(1 + 2x + [1 * x + 2x * x]) = x + 0.5
(1 + 2x + [1x + 2x2]) = x + 0.5

Combine like terms: 2x + 1x = 3x
(1 + 3x + 2x2) = x + 0.5

Reorder the terms:
1 + 3x + 2x2 = 0.5 + x

Solving
1 + 3x + 2x2 = 0.5 + x

Solving for variable 'x'.

Reorder the terms:
1 + -0.5 + 3x + -1x + 2x2 = 0.5 + x + -0.5 + -1x

Combine like terms: 1 + -0.5 = 0.5
0.5 + 3x + -1x + 2x2 = 0.5 + x + -0.5 + -1x

Combine like terms: 3x + -1x = 2x
0.5 + 2x + 2x2 = 0.5 + x + -0.5 + -1x

Reorder the terms:
0.5 + 2x + 2x2 = 0.5 + -0.5 + x + -1x

Combine like terms: 0.5 + -0.5 = 0.0
0.5 + 2x + 2x2 = 0.0 + x + -1x
0.5 + 2x + 2x2 = x + -1x

Combine like terms: x + -1x = 0
0.5 + 2x + 2x2 = 0

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

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

Move the constant term to the right:

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

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

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

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

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 = -0.25 + 0.25

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

Combine like terms: -0.25 + 0.25 = 0.00
1.25 + x2 = 0.00

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.00

Calculate the square root of the right side: 0.00

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.00 and 0.00.

Subproblem 1

x + 0.5 = 0.00 Simplifying x + 0.5 = 0.00 Reorder the terms: 0.5 + x = 0.00 Solving 0.5 + x = 0.00 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 = 0.00 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.00 + -0.5 x = 0.00 + -0.5 Combine like terms: 0.00 + -0.5 = -0.5 x = -0.5 Simplifying x = -0.5

Subproblem 2

x + 0.5 = 0.00 Simplifying x + 0.5 = 0.00 Reorder the terms: 0.5 + x = 0.00 Solving 0.5 + x = 0.00 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 = 0.00 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.00 + -0.5 x = 0.00 + -0.5 Combine like terms: 0.00 + -0.5 = -0.5 x = -0.5 Simplifying x = -0.5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.5, -0.5}

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