4x^2-6+2x^2+19x=180

Solution for 4x^2-6+2x^2+19x=180 equation:

Simplifying
4x2 + -6 + 2x2 + 19x = 180

Reorder the terms:
-6 + 19x + 4x2 + 2x2 = 180

Combine like terms: 4x2 + 2x2 = 6x2
-6 + 19x + 6x2 = 180

Solving
-6 + 19x + 6x2 = 180

Solving for variable 'x'.

Reorder the terms:
-6 + -180 + 19x + 6x2 = 180 + -180

Combine like terms: -6 + -180 = -186
-186 + 19x + 6x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-186 + 19x + 6x2 = 0

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

Divide each side by '6'.
-31 + 3.166666667x + x2 = 0

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + 3.166666667x + 31 + x2 = 0 + 31

Reorder the terms:
-31 + 31 + 3.166666667x + x2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + 3.166666667x + x2 = 0 + 31
3.166666667x + x2 = 0 + 31

Combine like terms: 0 + 31 = 31
3.166666667x + x2 = 31

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 = 31 + 2.506944447

Reorder the terms:
2.506944447 + 3.166666667x + x2 = 31 + 2.506944447

Combine like terms: 31 + 2.506944447 = 33.506944447
2.506944447 + 3.166666667x + x2 = 33.506944447

Factor a perfect square on the left side:
(x + 1.583333334)(x + 1.583333334) = 33.506944447

Calculate the square root of the right side: 5.788518329

Break this problem into two subproblems by setting 
(x + 1.583333334) equal to 5.788518329 and -5.788518329.

Subproblem 1
x + 1.583333334 = 5.788518329

Simplifying
x + 1.583333334 = 5.788518329

Reorder the terms:
1.583333334 + x = 5.788518329

Solving
1.583333334 + x = 5.788518329

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 = 5.788518329 + -1.583333334

Combine like terms: 1.583333334 + -1.583333334 = 0.000000000
0.000000000 + x = 5.788518329 + -1.583333334
x = 5.788518329 + -1.583333334

Combine like terms: 5.788518329 + -1.583333334 = 4.205184995
x = 4.205184995

Simplifying
x = 4.205184995

Subproblem 2
x + 1.583333334 = -5.788518329

Simplifying
x + 1.583333334 = -5.788518329

Reorder the terms:
1.583333334 + x = -5.788518329

Solving
1.583333334 + x = -5.788518329

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 = -5.788518329 + -1.583333334

Combine like terms: 1.583333334 + -1.583333334 = 0.000000000
0.000000000 + x = -5.788518329 + -1.583333334
x = -5.788518329 + -1.583333334

Combine like terms: -5.788518329 + -1.583333334 = -7.371851663
x = -7.371851663

Simplifying
x = -7.371851663

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.205184995, -7.371851663}