5.1=x^2+3x+1

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Solution for 5.1=x^2+3x+1 equation: 


Simplifying
5.1 = x2 + 3x + 1

Reorder the terms:
5.1 = 1 + 3x + x2

Solving
5.1 = 1 + 3x + x2

Solving for variable 'x'.

Combine like terms: 5.1 + -1 = 4.1
4.1 + -3x + -1x2 = 1 + 3x + x2 + -1 + -3x + -1x2

Reorder the terms:
4.1 + -3x + -1x2 = 1 + -1 + 3x + -3x + x2 + -1x2

Combine like terms: 1 + -1 = 0
4.1 + -3x + -1x2 = 0 + 3x + -3x + x2 + -1x2
4.1 + -3x + -1x2 = 3x + -3x + x2 + -1x2

Combine like terms: 3x + -3x = 0
4.1 + -3x + -1x2 = 0 + x2 + -1x2
4.1 + -3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
4.1 + -3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-4.1 + 3x + x2 = 0

Move the constant term to the right:

Add '4.1' to each side of the equation.
-4.1 + 3x + 4.1 + x2 = 0 + 4.1

Reorder the terms:
-4.1 + 4.1 + 3x + x2 = 0 + 4.1

Combine like terms: -4.1 + 4.1 = 0.0
0.0 + 3x + x2 = 0 + 4.1
3x + x2 = 0 + 4.1

Combine like terms: 0 + 4.1 = 4.1
3x + x2 = 4.1

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 = 4.1 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 4.1 + 2.25

Combine like terms: 4.1 + 2.25 = 6.35
2.25 + 3x + x2 = 6.35

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 6.35

Calculate the square root of the right side: 2.519920634

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.519920634 and -2.519920634.


Subproblem 1
x + 1.5 = 2.519920634

Simplifying
x + 1.5 = 2.519920634

Reorder the terms:
1.5 + x = 2.519920634

Solving
1.5 + x = 2.519920634

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 = 2.519920634 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 2.519920634 + -1.5
x = 2.519920634 + -1.5

Combine like terms: 2.519920634 + -1.5 = 1.019920634
x = 1.019920634

Simplifying
x = 1.019920634


Subproblem 2
x + 1.5 = -2.519920634

Simplifying
x + 1.5 = -2.519920634

Reorder the terms:
1.5 + x = -2.519920634

Solving
1.5 + x = -2.519920634

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 = -2.519920634 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -2.519920634 + -1.5
x = -2.519920634 + -1.5

Combine like terms: -2.519920634 + -1.5 = -4.019920634
x = -4.019920634

Simplifying
x = -4.019920634


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.019920634, -4.019920634}

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