x^2+3x-4.5=0

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Solution for x^2+3x-4.5=0 equation: 


Simplifying
x2 + 3x + -4.5 = 0

Reorder the terms:
-4.5 + 3x + x2 = 0

Solving
-4.5 + 3x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 3x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 3x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 3x + x2 = 0 + 4.5
3x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
3x + x2 = 4.5

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 = 4.5 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 4.5 + 2.25

Combine like terms: 4.5 + 2.25 = 6.75
2.25 + 3x + x2 = 6.75

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.598076211 and -2.598076211.


Subproblem 1
x + 1.5 = 2.598076211

Simplifying
x + 1.5 = 2.598076211

Reorder the terms:
1.5 + x = 2.598076211

Solving
1.5 + x = 2.598076211

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 = 2.598076211 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 2.598076211 + -1.5
x = 2.598076211 + -1.5

Combine like terms: 2.598076211 + -1.5 = 1.098076211
x = 1.098076211

Simplifying
x = 1.098076211


Subproblem 2
x + 1.5 = -2.598076211

Simplifying
x + 1.5 = -2.598076211

Reorder the terms:
1.5 + x = -2.598076211

Solving
1.5 + x = -2.598076211

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 = -2.598076211 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -2.598076211 + -1.5
x = -2.598076211 + -1.5

Combine like terms: -2.598076211 + -1.5 = -4.098076211
x = -4.098076211

Simplifying
x = -4.098076211


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.098076211, -4.098076211}

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