6x^2+22x+19=0

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Solution for 6x^2+22x+19=0 equation:


Simplifying
6x2 + 22x + 19 = 0

Reorder the terms:
19 + 22x + 6x2 = 0

Solving
19 + 22x + 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'.
3.166666667 + 3.666666667x + x2 = 0

Move the constant term to the right:

Add '-3.166666667' to each side of the equation.
3.166666667 + 3.666666667x + -3.166666667 + x2 = 0 + -3.166666667

Reorder the terms:
3.166666667 + -3.166666667 + 3.666666667x + x2 = 0 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + 3.666666667x + x2 = 0 + -3.166666667
3.666666667x + x2 = 0 + -3.166666667

Combine like terms: 0 + -3.166666667 = -3.166666667
3.666666667x + x2 = -3.166666667

The x term is 3.666666667x.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667x + 3.361111114 + x2 = -3.166666667 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667x + x2 = -3.166666667 + 3.361111114

Combine like terms: -3.166666667 + 3.361111114 = 0.194444447
3.361111114 + 3.666666667x + x2 = 0.194444447

Factor a perfect square on the left side:
(x + 1.833333334)(x + 1.833333334) = 0.194444447

Calculate the square root of the right side: 0.440958555

Break this problem into two subproblems by setting 
(x + 1.833333334) equal to 0.440958555 and -0.440958555.

Subproblem 1

x + 1.833333334 = 0.440958555 Simplifying x + 1.833333334 = 0.440958555 Reorder the terms: 1.833333334 + x = 0.440958555 Solving 1.833333334 + x = 0.440958555 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = 0.440958555 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 0.440958555 + -1.833333334 x = 0.440958555 + -1.833333334 Combine like terms: 0.440958555 + -1.833333334 = -1.392374779 x = -1.392374779 Simplifying x = -1.392374779

Subproblem 2

x + 1.833333334 = -0.440958555 Simplifying x + 1.833333334 = -0.440958555 Reorder the terms: 1.833333334 + x = -0.440958555 Solving 1.833333334 + x = -0.440958555 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = -0.440958555 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -0.440958555 + -1.833333334 x = -0.440958555 + -1.833333334 Combine like terms: -0.440958555 + -1.833333334 = -2.274291889 x = -2.274291889 Simplifying x = -2.274291889

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.392374779, -2.274291889}

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