6x^2+19x+4=0

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Solution for 6x^2+19x+4=0 equation:


Simplifying
6x2 + 19x + 4 = 0

Reorder the terms:
4 + 19x + 6x2 = 0

Solving
4 + 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'.
0.6666666667 + 3.166666667x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + 3.166666667x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + 3.166666667x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + 3.166666667x + x2 = 0 + -0.6666666667
3.166666667x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
3.166666667x + x2 = -0.6666666667

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 = -0.6666666667 + 2.506944447

Reorder the terms:
2.506944447 + 3.166666667x + x2 = -0.6666666667 + 2.506944447

Combine like terms: -0.6666666667 + 2.506944447 = 1.8402777803
2.506944447 + 3.166666667x + x2 = 1.8402777803

Factor a perfect square on the left side:
(x + 1.583333334)(x + 1.583333334) = 1.8402777803

Calculate the square root of the right side: 1.356568384

Break this problem into two subproblems by setting 
(x + 1.583333334) equal to 1.356568384 and -1.356568384.

Subproblem 1

x + 1.583333334 = 1.356568384 Simplifying x + 1.583333334 = 1.356568384 Reorder the terms: 1.583333334 + x = 1.356568384 Solving 1.583333334 + x = 1.356568384 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 = 1.356568384 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = 1.356568384 + -1.583333334 x = 1.356568384 + -1.583333334 Combine like terms: 1.356568384 + -1.583333334 = -0.22676495 x = -0.22676495 Simplifying x = -0.22676495

Subproblem 2

x + 1.583333334 = -1.356568384 Simplifying x + 1.583333334 = -1.356568384 Reorder the terms: 1.583333334 + x = -1.356568384 Solving 1.583333334 + x = -1.356568384 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 = -1.356568384 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = -1.356568384 + -1.583333334 x = -1.356568384 + -1.583333334 Combine like terms: -1.356568384 + -1.583333334 = -2.939901718 x = -2.939901718 Simplifying x = -2.939901718

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.22676495, -2.939901718}

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