2x^2+4x=x^2+x+1

Solution for 2x^2+4x=x^2+x+1 equation:

Simplifying
2x2 + 4x = x2 + x + 1

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

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

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

Solving for variable 'x'.

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

Combine like terms: 4x + -1x = 3x
-1 + 3x + 2x2 + -1x2 = 1 + x + x2 + -1 + -1x + -1x2

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

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

Combine like terms: 1 + -1 = 0
-1 + 3x + 1x2 = 0 + x + -1x + x2 + -1x2
-1 + 3x + 1x2 = x + -1x + x2 + -1x2

Combine like terms: x + -1x = 0
-1 + 3x + 1x2 = 0 + x2 + -1x2
-1 + 3x + 1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-1 + 3x + 1x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 3x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 3x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 3x + x2 = 0 + 1
3x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
3x + x2 = 1

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 1 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 1 + 2.25

Combine like terms: 1 + 2.25 = 3.25
2.25 + 3x + x2 = 3.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(x + 1.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
x + 1.5 = 1.802775638

Simplifying
x + 1.5 = 1.802775638

Reorder the terms:
1.5 + x = 1.802775638

Solving
1.5 + x = 1.802775638

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 1.802775638 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 1.802775638 + -1.5
x = 1.802775638 + -1.5

Combine like terms: 1.802775638 + -1.5 = 0.302775638
x = 0.302775638

Simplifying
x = 0.302775638

Subproblem 2
x + 1.5 = -1.802775638

Simplifying
x + 1.5 = -1.802775638

Reorder the terms:
1.5 + x = -1.802775638

Solving
1.5 + x = -1.802775638

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -1.802775638 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -1.802775638 + -1.5
x = -1.802775638 + -1.5

Combine like terms: -1.802775638 + -1.5 = -3.302775638
x = -3.302775638

Simplifying
x = -3.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.302775638, -3.302775638}