6x^2+19x+12=0

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Solution for 6x^2+19x+12=0 equation:


Simplifying
6x2 + 19x + 12 = 0

Reorder the terms:
12 + 19x + 6x2 = 0

Solving
12 + 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'.
2 + 3.166666667x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 3.166666667x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + 3.166666667x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 3.166666667x + x2 = 0 + -2
3.166666667x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
3.166666667x + x2 = -2

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 = -2 + 2.506944447

Reorder the terms:
2.506944447 + 3.166666667x + x2 = -2 + 2.506944447

Combine like terms: -2 + 2.506944447 = 0.506944447
2.506944447 + 3.166666667x + x2 = 0.506944447

Factor a perfect square on the left side:
(x + 1.583333334)(x + 1.583333334) = 0.506944447

Calculate the square root of the right side: 0.712000314

Break this problem into two subproblems by setting 
(x + 1.583333334) equal to 0.712000314 and -0.712000314.

Subproblem 1

x + 1.583333334 = 0.712000314 Simplifying x + 1.583333334 = 0.712000314 Reorder the terms: 1.583333334 + x = 0.712000314 Solving 1.583333334 + x = 0.712000314 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 = 0.712000314 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = 0.712000314 + -1.583333334 x = 0.712000314 + -1.583333334 Combine like terms: 0.712000314 + -1.583333334 = -0.87133302 x = -0.87133302 Simplifying x = -0.87133302

Subproblem 2

x + 1.583333334 = -0.712000314 Simplifying x + 1.583333334 = -0.712000314 Reorder the terms: 1.583333334 + x = -0.712000314 Solving 1.583333334 + x = -0.712000314 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 = -0.712000314 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = -0.712000314 + -1.583333334 x = -0.712000314 + -1.583333334 Combine like terms: -0.712000314 + -1.583333334 = -2.295333648 x = -2.295333648 Simplifying x = -2.295333648

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.87133302, -2.295333648}

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