-3x^2-5x+19=0

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Solution for -3x^2-5x+19=0 equation: 


Simplifying
-3x2 + -5x + 19 = 0

Reorder the terms:
19 + -5x + -3x2 = 0

Solving
19 + -5x + -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'.
-6.333333333 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '6.333333333' to each side of the equation.
-6.333333333 + 1.666666667x + 6.333333333 + x2 = 0 + 6.333333333

Reorder the terms:
-6.333333333 + 6.333333333 + 1.666666667x + x2 = 0 + 6.333333333

Combine like terms: -6.333333333 + 6.333333333 = 0.000000000
0.000000000 + 1.666666667x + x2 = 0 + 6.333333333
1.666666667x + x2 = 0 + 6.333333333

Combine like terms: 0 + 6.333333333 = 6.333333333
1.666666667x + x2 = 6.333333333

The x term is 1.666666667x.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667x + 0.6944444447 + x2 = 6.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 6.333333333 + 0.6944444447

Combine like terms: 6.333333333 + 0.6944444447 = 7.0277777777
0.6944444447 + 1.666666667x + x2 = 7.0277777777

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 7.0277777777

Calculate the square root of the right side: 2.65099562

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 2.65099562 and -2.65099562.


Subproblem 1
x + 0.8333333335 = 2.65099562

Simplifying
x + 0.8333333335 = 2.65099562

Reorder the terms:
0.8333333335 + x = 2.65099562

Solving
0.8333333335 + x = 2.65099562

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = 2.65099562 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = 2.65099562 + -0.8333333335
x = 2.65099562 + -0.8333333335

Combine like terms: 2.65099562 + -0.8333333335 = 1.8176622865
x = 1.8176622865

Simplifying
x = 1.8176622865


Subproblem 2
x + 0.8333333335 = -2.65099562

Simplifying
x + 0.8333333335 = -2.65099562

Reorder the terms:
0.8333333335 + x = -2.65099562

Solving
0.8333333335 + x = -2.65099562

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = -2.65099562 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = -2.65099562 + -0.8333333335
x = -2.65099562 + -0.8333333335

Combine like terms: -2.65099562 + -0.8333333335 = -3.4843289535
x = -3.4843289535

Simplifying
x = -3.4843289535


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.8176622865, -3.4843289535}

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