3x^2+19x-6=0

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Solution for 3x^2+19x-6=0 equation:


Simplifying
3x2 + 19x + -6 = 0

Reorder the terms:
-6 + 19x + 3x2 = 0

Solving
-6 + 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'.
-2 + 6.333333333x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 6.333333333x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + 6.333333333x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 6.333333333x + x2 = 0 + 2
6.333333333x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
6.333333333x + x2 = 2

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 = 2 + 10.02777778

Reorder the terms:
10.02777778 + 6.333333333x + x2 = 2 + 10.02777778

Combine like terms: 2 + 10.02777778 = 12.02777778
10.02777778 + 6.333333333x + x2 = 12.02777778

Factor a perfect square on the left side:
(x + 3.166666667)(x + 3.166666667) = 12.02777778

Calculate the square root of the right side: 3.468108675

Break this problem into two subproblems by setting 
(x + 3.166666667) equal to 3.468108675 and -3.468108675.

Subproblem 1

x + 3.166666667 = 3.468108675 Simplifying x + 3.166666667 = 3.468108675 Reorder the terms: 3.166666667 + x = 3.468108675 Solving 3.166666667 + x = 3.468108675 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 = 3.468108675 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = 3.468108675 + -3.166666667 x = 3.468108675 + -3.166666667 Combine like terms: 3.468108675 + -3.166666667 = 0.301442008 x = 0.301442008 Simplifying x = 0.301442008

Subproblem 2

x + 3.166666667 = -3.468108675 Simplifying x + 3.166666667 = -3.468108675 Reorder the terms: 3.166666667 + x = -3.468108675 Solving 3.166666667 + x = -3.468108675 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 = -3.468108675 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = -3.468108675 + -3.166666667 x = -3.468108675 + -3.166666667 Combine like terms: -3.468108675 + -3.166666667 = -6.634775342 x = -6.634775342 Simplifying x = -6.634775342

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.301442008, -6.634775342}

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