6x^2+19x-6=0

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


Simplifying
6x2 + 19x + -6 = 0

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

Solving
-6 + 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'.
-1 + 3.166666667x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 3.166666667x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 3.166666667x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 3.166666667x + x2 = 0 + 1
3.166666667x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
3.166666667x + x2 = 1

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 = 1 + 2.506944447

Reorder the terms:
2.506944447 + 3.166666667x + x2 = 1 + 2.506944447

Combine like terms: 1 + 2.506944447 = 3.506944447
2.506944447 + 3.166666667x + x2 = 3.506944447

Factor a perfect square on the left side:
(x + 1.583333334)(x + 1.583333334) = 3.506944447

Calculate the square root of the right side: 1.872683755

Break this problem into two subproblems by setting 
(x + 1.583333334) equal to 1.872683755 and -1.872683755.

Subproblem 1

x + 1.583333334 = 1.872683755 Simplifying x + 1.583333334 = 1.872683755 Reorder the terms: 1.583333334 + x = 1.872683755 Solving 1.583333334 + x = 1.872683755 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 = 1.872683755 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = 1.872683755 + -1.583333334 x = 1.872683755 + -1.583333334 Combine like terms: 1.872683755 + -1.583333334 = 0.289350421 x = 0.289350421 Simplifying x = 0.289350421

Subproblem 2

x + 1.583333334 = -1.872683755 Simplifying x + 1.583333334 = -1.872683755 Reorder the terms: 1.583333334 + x = -1.872683755 Solving 1.583333334 + x = -1.872683755 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 = -1.872683755 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = -1.872683755 + -1.583333334 x = -1.872683755 + -1.583333334 Combine like terms: -1.872683755 + -1.583333334 = -3.456017089 x = -3.456017089 Simplifying x = -3.456017089

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.289350421, -3.456017089}

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