x^2-3x-1=0.5

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Solution for x^2-3x-1=0.5 equation: 


Simplifying
x2 + -3x + -1 = 0.5

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

Solving
-1 + -3x + x2 = 0.5

Solving for variable 'x'.

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

Combine like terms: -1 + -0.5 = -1.5
-1.5 + -3x + x2 = 0.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
-1.5 + -3x + x2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -3x + 1.5 + x2 = 0.0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -3x + x2 = 0.0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -3x + x2 = 0.0 + 1.5
-3x + x2 = 0.0 + 1.5

Combine like terms: 0.0 + 1.5 = 1.5
-3x + x2 = 1.5

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 = 1.5 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 1.5 + 2.25

Combine like terms: 1.5 + 2.25 = 3.75
2.25 + -3x + x2 = 3.75

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.936491673 and -1.936491673.


Subproblem 1
x + -1.5 = 1.936491673

Simplifying
x + -1.5 = 1.936491673

Reorder the terms:
-1.5 + x = 1.936491673

Solving
-1.5 + x = 1.936491673

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 = 1.936491673 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 1.936491673 + 1.5
x = 1.936491673 + 1.5

Combine like terms: 1.936491673 + 1.5 = 3.436491673
x = 3.436491673

Simplifying
x = 3.436491673


Subproblem 2
x + -1.5 = -1.936491673

Simplifying
x + -1.5 = -1.936491673

Reorder the terms:
-1.5 + x = -1.936491673

Solving
-1.5 + x = -1.936491673

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 = -1.936491673 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -1.936491673 + 1.5
x = -1.936491673 + 1.5

Combine like terms: -1.936491673 + 1.5 = -0.436491673
x = -0.436491673

Simplifying
x = -0.436491673


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.436491673, -0.436491673}

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