6x^2+19x-120=0

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Solution for 6x^2+19x-120=0 equation: 


Simplifying
6x2 + 19x + -120 = 0

Reorder the terms:
-120 + 19x + 6x2 = 0

Solving
-120 + 19x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-20 + 3.166666667x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 3.166666667x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + 3.166666667x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 3.166666667x + x2 = 0 + 20
3.166666667x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
3.166666667x + x2 = 20

The x term is 3.166666667x.  Take half its coefficient (1.583333334).
Square it (2.506944447) and add it to both sides.

Add '2.506944447' to each side of the equation.
3.166666667x + 2.506944447 + x2 = 20 + 2.506944447

Reorder the terms:
2.506944447 + 3.166666667x + x2 = 20 + 2.506944447

Combine like terms: 20 + 2.506944447 = 22.506944447
2.506944447 + 3.166666667x + x2 = 22.506944447

Factor a perfect square on the left side:
(x + 1.583333334)(x + 1.583333334) = 22.506944447

Calculate the square root of the right side: 4.744148443

Break this problem into two subproblems by setting 
(x + 1.583333334) equal to 4.744148443 and -4.744148443.


Subproblem 1
x + 1.583333334 = 4.744148443

Simplifying
x + 1.583333334 = 4.744148443

Reorder the terms:
1.583333334 + x = 4.744148443

Solving
1.583333334 + x = 4.744148443

Solving for variable 'x'.

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

Add '-1.583333334' to each side of the equation.
1.583333334 + -1.583333334 + x = 4.744148443 + -1.583333334

Combine like terms: 1.583333334 + -1.583333334 = 0.000000000
0.000000000 + x = 4.744148443 + -1.583333334
x = 4.744148443 + -1.583333334

Combine like terms: 4.744148443 + -1.583333334 = 3.160815109
x = 3.160815109

Simplifying
x = 3.160815109


Subproblem 2
x + 1.583333334 = -4.744148443

Simplifying
x + 1.583333334 = -4.744148443

Reorder the terms:
1.583333334 + x = -4.744148443

Solving
1.583333334 + x = -4.744148443

Solving for variable 'x'.

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

Add '-1.583333334' to each side of the equation.
1.583333334 + -1.583333334 + x = -4.744148443 + -1.583333334

Combine like terms: 1.583333334 + -1.583333334 = 0.000000000
0.000000000 + x = -4.744148443 + -1.583333334
x = -4.744148443 + -1.583333334

Combine like terms: -4.744148443 + -1.583333334 = -6.327481777
x = -6.327481777

Simplifying
x = -6.327481777


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.160815109, -6.327481777}

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