x+(x+1)=(2x-1)(x+4)

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Solution for x+(x+1)=(2x-1)(x+4) equation:


Simplifying
x + (x + 1) = (2x + -1)(x + 4)

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

Remove parenthesis around (1 + x)
x + 1 + x = (2x + -1)(x + 4)

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

Combine like terms: x + x = 2x
1 + 2x = (2x + -1)(x + 4)

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

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

Multiply (-1 + 2x) * (4 + x)
1 + 2x = (-1(4 + x) + 2x * (4 + x))
1 + 2x = ((4 * -1 + x * -1) + 2x * (4 + x))
1 + 2x = ((-4 + -1x) + 2x * (4 + x))
1 + 2x = (-4 + -1x + (4 * 2x + x * 2x))
1 + 2x = (-4 + -1x + (8x + 2x2))

Combine like terms: -1x + 8x = 7x
1 + 2x = (-4 + 7x + 2x2)

Solving
1 + 2x = -4 + 7x + 2x2

Solving for variable 'x'.

Reorder the terms:
1 + 4 + 2x + -7x + -2x2 = -4 + 7x + 2x2 + 4 + -7x + -2x2

Combine like terms: 1 + 4 = 5
5 + 2x + -7x + -2x2 = -4 + 7x + 2x2 + 4 + -7x + -2x2

Combine like terms: 2x + -7x = -5x
5 + -5x + -2x2 = -4 + 7x + 2x2 + 4 + -7x + -2x2

Reorder the terms:
5 + -5x + -2x2 = -4 + 4 + 7x + -7x + 2x2 + -2x2

Combine like terms: -4 + 4 = 0
5 + -5x + -2x2 = 0 + 7x + -7x + 2x2 + -2x2
5 + -5x + -2x2 = 7x + -7x + 2x2 + -2x2

Combine like terms: 7x + -7x = 0
5 + -5x + -2x2 = 0 + 2x2 + -2x2
5 + -5x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
5 + -5x + -2x2 = 0

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

Divide each side by '-2'.
-2.5 + 2.5x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 2.5x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 2.5x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 2.5x + x2 = 0 + 2.5
2.5x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
2.5x + x2 = 2.5

The x term is 2.5x.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x + 1.5625 + x2 = 2.5 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = 2.5 + 1.5625

Combine like terms: 2.5 + 1.5625 = 4.0625
1.5625 + 2.5x + x2 = 4.0625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(x + 1.25) equal to 2.015564437 and -2.015564437.

Subproblem 1

x + 1.25 = 2.015564437 Simplifying x + 1.25 = 2.015564437 Reorder the terms: 1.25 + x = 2.015564437 Solving 1.25 + x = 2.015564437 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 2.015564437 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 2.015564437 + -1.25 x = 2.015564437 + -1.25 Combine like terms: 2.015564437 + -1.25 = 0.765564437 x = 0.765564437 Simplifying x = 0.765564437

Subproblem 2

x + 1.25 = -2.015564437 Simplifying x + 1.25 = -2.015564437 Reorder the terms: 1.25 + x = -2.015564437 Solving 1.25 + x = -2.015564437 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -2.015564437 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -2.015564437 + -1.25 x = -2.015564437 + -1.25 Combine like terms: -2.015564437 + -1.25 = -3.265564437 x = -3.265564437 Simplifying x = -3.265564437

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.765564437, -3.265564437}

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