19.416x^2+108.672x+125.78=0

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Solution for 19.416x^2+108.672x+125.78=0 equation: 


Simplifying
19.416x2 + 108.672x + 125.78 = 0

Reorder the terms:
125.78 + 108.672x + 19.416x2 = 0

Solving
125.78 + 108.672x + 19.416x2 = 0

Solving for variable 'x'.

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

Divide each side by '19.416'.
6.47816234 + 5.597033375x + x2 = 0

Move the constant term to the right:

Add '-6.47816234' to each side of the equation.
6.47816234 + 5.597033375x + -6.47816234 + x2 = 0 + -6.47816234

Reorder the terms:
6.47816234 + -6.47816234 + 5.597033375x + x2 = 0 + -6.47816234

Combine like terms: 6.47816234 + -6.47816234 = 0.00000000
0.00000000 + 5.597033375x + x2 = 0 + -6.47816234
5.597033375x + x2 = 0 + -6.47816234

Combine like terms: 0 + -6.47816234 = -6.47816234
5.597033375x + x2 = -6.47816234

The x term is 5.597033375x.  Take half its coefficient (2.798516688).
Square it (7.831695653) and add it to both sides.

Add '7.831695653' to each side of the equation.
5.597033375x + 7.831695653 + x2 = -6.47816234 + 7.831695653

Reorder the terms:
7.831695653 + 5.597033375x + x2 = -6.47816234 + 7.831695653

Combine like terms: -6.47816234 + 7.831695653 = 1.353533313
7.831695653 + 5.597033375x + x2 = 1.353533313

Factor a perfect square on the left side:
(x + 2.798516688)(x + 2.798516688) = 1.353533313

Calculate the square root of the right side: 1.163414506

Break this problem into two subproblems by setting 
(x + 2.798516688) equal to 1.163414506 and -1.163414506.


Subproblem 1
x + 2.798516688 = 1.163414506

Simplifying
x + 2.798516688 = 1.163414506

Reorder the terms:
2.798516688 + x = 1.163414506

Solving
2.798516688 + x = 1.163414506

Solving for variable 'x'.

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

Add '-2.798516688' to each side of the equation.
2.798516688 + -2.798516688 + x = 1.163414506 + -2.798516688

Combine like terms: 2.798516688 + -2.798516688 = 0.000000000
0.000000000 + x = 1.163414506 + -2.798516688
x = 1.163414506 + -2.798516688

Combine like terms: 1.163414506 + -2.798516688 = -1.635102182
x = -1.635102182

Simplifying
x = -1.635102182


Subproblem 2
x + 2.798516688 = -1.163414506

Simplifying
x + 2.798516688 = -1.163414506

Reorder the terms:
2.798516688 + x = -1.163414506

Solving
2.798516688 + x = -1.163414506

Solving for variable 'x'.

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

Add '-2.798516688' to each side of the equation.
2.798516688 + -2.798516688 + x = -1.163414506 + -2.798516688

Combine like terms: 2.798516688 + -2.798516688 = 0.000000000
0.000000000 + x = -1.163414506 + -2.798516688
x = -1.163414506 + -2.798516688

Combine like terms: -1.163414506 + -2.798516688 = -3.961931194
x = -3.961931194

Simplifying
x = -3.961931194


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.635102182, -3.961931194}

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