3x^2-5x-13=0

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Solution for 3x^2-5x-13=0 equation: 


Simplifying
3x2 + -5x + -13 = 0

Reorder the terms:
-13 + -5x + 3x2 = 0

Solving
-13 + -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'.
-4.333333333 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '4.333333333' to each side of the equation.
-4.333333333 + -1.666666667x + 4.333333333 + x2 = 0 + 4.333333333

Reorder the terms:
-4.333333333 + 4.333333333 + -1.666666667x + x2 = 0 + 4.333333333

Combine like terms: -4.333333333 + 4.333333333 = 0.000000000
0.000000000 + -1.666666667x + x2 = 0 + 4.333333333
-1.666666667x + x2 = 0 + 4.333333333

Combine like terms: 0 + 4.333333333 = 4.333333333
-1.666666667x + x2 = 4.333333333

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 = 4.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 4.333333333 + 0.6944444447

Combine like terms: 4.333333333 + 0.6944444447 = 5.0277777777
0.6944444447 + -1.666666667x + x2 = 5.0277777777

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 5.0277777777

Calculate the square root of the right side: 2.242270674

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 2.242270674 and -2.242270674.


Subproblem 1
x + -0.8333333335 = 2.242270674

Simplifying
x + -0.8333333335 = 2.242270674

Reorder the terms:
-0.8333333335 + x = 2.242270674

Solving
-0.8333333335 + x = 2.242270674

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 = 2.242270674 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = 2.242270674 + 0.8333333335
x = 2.242270674 + 0.8333333335

Combine like terms: 2.242270674 + 0.8333333335 = 3.0756040075
x = 3.0756040075

Simplifying
x = 3.0756040075


Subproblem 2
x + -0.8333333335 = -2.242270674

Simplifying
x + -0.8333333335 = -2.242270674

Reorder the terms:
-0.8333333335 + x = -2.242270674

Solving
-0.8333333335 + x = -2.242270674

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 = -2.242270674 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = -2.242270674 + 0.8333333335
x = -2.242270674 + 0.8333333335

Combine like terms: -2.242270674 + 0.8333333335 = -1.4089373405
x = -1.4089373405

Simplifying
x = -1.4089373405


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.0756040075, -1.4089373405}

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