3x^2+19x+4=0

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Solution for 3x^2+19x+4=0 equation: 


Simplifying
3x2 + 19x + 4 = 0

Reorder the terms:
4 + 19x + 3x2 = 0

Solving
4 + 19x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
1.333333333 + 6.333333333x + x2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + 6.333333333x + -1.333333333 + x2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + 6.333333333x + x2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + 6.333333333x + x2 = 0 + -1.333333333
6.333333333x + x2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
6.333333333x + x2 = -1.333333333

The x term is 6.333333333x.  Take half its coefficient (3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
6.333333333x + 10.02777778 + x2 = -1.333333333 + 10.02777778

Reorder the terms:
10.02777778 + 6.333333333x + x2 = -1.333333333 + 10.02777778

Combine like terms: -1.333333333 + 10.02777778 = 8.694444447
10.02777778 + 6.333333333x + x2 = 8.694444447

Factor a perfect square on the left side:
(x + 3.166666667)(x + 3.166666667) = 8.694444447

Calculate the square root of the right side: 2.948634336

Break this problem into two subproblems by setting 
(x + 3.166666667) equal to 2.948634336 and -2.948634336.


Subproblem 1
x + 3.166666667 = 2.948634336

Simplifying
x + 3.166666667 = 2.948634336

Reorder the terms:
3.166666667 + x = 2.948634336

Solving
3.166666667 + x = 2.948634336

Solving for variable 'x'.

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

Add '-3.166666667' to each side of the equation.
3.166666667 + -3.166666667 + x = 2.948634336 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + x = 2.948634336 + -3.166666667
x = 2.948634336 + -3.166666667

Combine like terms: 2.948634336 + -3.166666667 = -0.218032331
x = -0.218032331

Simplifying
x = -0.218032331


Subproblem 2
x + 3.166666667 = -2.948634336

Simplifying
x + 3.166666667 = -2.948634336

Reorder the terms:
3.166666667 + x = -2.948634336

Solving
3.166666667 + x = -2.948634336

Solving for variable 'x'.

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

Add '-3.166666667' to each side of the equation.
3.166666667 + -3.166666667 + x = -2.948634336 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + x = -2.948634336 + -3.166666667
x = -2.948634336 + -3.166666667

Combine like terms: -2.948634336 + -3.166666667 = -6.115301003
x = -6.115301003

Simplifying
x = -6.115301003


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.218032331, -6.115301003}

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