x^2+x+1=4.5

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Solution for x^2+x+1=4.5 equation: 


Simplifying
x2 + x + 1 = 4.5

Reorder the terms:
1 + x + x2 = 4.5

Solving
1 + x + x2 = 4.5

Solving for variable 'x'.

Reorder the terms:
1 + -4.5 + x + x2 = 4.5 + -4.5

Combine like terms: 1 + -4.5 = -3.5
-3.5 + x + x2 = 4.5 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
-3.5 + x + x2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + x + 3.5 + x2 = 0.0 + 3.5

Reorder the terms:
-3.5 + 3.5 + x + x2 = 0.0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x + x2 = 0.0 + 3.5
x + x2 = 0.0 + 3.5

Combine like terms: 0.0 + 3.5 = 3.5
x + x2 = 3.5

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.
x + 0.25 + x2 = 3.5 + 0.25

Reorder the terms:
0.25 + x + x2 = 3.5 + 0.25

Combine like terms: 3.5 + 0.25 = 3.75
0.25 + x + x2 = 3.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.936491673 and -1.936491673.


Subproblem 1
x + 0.5 = 1.936491673

Simplifying
x + 0.5 = 1.936491673

Reorder the terms:
0.5 + x = 1.936491673

Solving
0.5 + x = 1.936491673

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 = 1.936491673 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 1.936491673 + -0.5
x = 1.936491673 + -0.5

Combine like terms: 1.936491673 + -0.5 = 1.436491673
x = 1.436491673

Simplifying
x = 1.436491673


Subproblem 2
x + 0.5 = -1.936491673

Simplifying
x + 0.5 = -1.936491673

Reorder the terms:
0.5 + x = -1.936491673

Solving
0.5 + x = -1.936491673

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 = -1.936491673 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -1.936491673 + -0.5
x = -1.936491673 + -0.5

Combine like terms: -1.936491673 + -0.5 = -2.436491673
x = -2.436491673

Simplifying
x = -2.436491673


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.436491673, -2.436491673}

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