6x^2+25x-19=0

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Solution for 6x^2+25x-19=0 equation: 


Simplifying
6x2 + 25x + -19 = 0

Reorder the terms:
-19 + 25x + 6x2 = 0

Solving
-19 + 25x + 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 + 4.166666667x + x2 = 0

Move the constant term to the right:

Add '3.166666667' to each side of the equation.
-3.166666667 + 4.166666667x + 3.166666667 + x2 = 0 + 3.166666667

Reorder the terms:
-3.166666667 + 3.166666667 + 4.166666667x + x2 = 0 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + 4.166666667x + x2 = 0 + 3.166666667
4.166666667x + x2 = 0 + 3.166666667

Combine like terms: 0 + 3.166666667 = 3.166666667
4.166666667x + x2 = 3.166666667

The x term is 4.166666667x.  Take half its coefficient (2.083333334).
Square it (4.340277781) and add it to both sides.

Add '4.340277781' to each side of the equation.
4.166666667x + 4.340277781 + x2 = 3.166666667 + 4.340277781

Reorder the terms:
4.340277781 + 4.166666667x + x2 = 3.166666667 + 4.340277781

Combine like terms: 3.166666667 + 4.340277781 = 7.506944448
4.340277781 + 4.166666667x + x2 = 7.506944448

Factor a perfect square on the left side:
(x + 2.083333334)(x + 2.083333334) = 7.506944448

Calculate the square root of the right side: 2.739880371

Break this problem into two subproblems by setting 
(x + 2.083333334) equal to 2.739880371 and -2.739880371.


Subproblem 1
x + 2.083333334 = 2.739880371

Simplifying
x + 2.083333334 = 2.739880371

Reorder the terms:
2.083333334 + x = 2.739880371

Solving
2.083333334 + x = 2.739880371

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.083333334' to each side of the equation.
2.083333334 + -2.083333334 + x = 2.739880371 + -2.083333334

Combine like terms: 2.083333334 + -2.083333334 = 0.000000000
0.000000000 + x = 2.739880371 + -2.083333334
x = 2.739880371 + -2.083333334

Combine like terms: 2.739880371 + -2.083333334 = 0.656547037
x = 0.656547037

Simplifying
x = 0.656547037


Subproblem 2
x + 2.083333334 = -2.739880371

Simplifying
x + 2.083333334 = -2.739880371

Reorder the terms:
2.083333334 + x = -2.739880371

Solving
2.083333334 + x = -2.739880371

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.083333334' to each side of the equation.
2.083333334 + -2.083333334 + x = -2.739880371 + -2.083333334

Combine like terms: 2.083333334 + -2.083333334 = 0.000000000
0.000000000 + x = -2.739880371 + -2.083333334
x = -2.739880371 + -2.083333334

Combine like terms: -2.739880371 + -2.083333334 = -4.823213705
x = -4.823213705

Simplifying
x = -4.823213705


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.656547037, -4.823213705}

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