0=3x^2-5x-13

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


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

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

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

Solving for variable 'x'.

Combine like terms: 0 + 13 = 13
13 + 5x + -3x2 = -13 + -5x + 3x2 + 13 + 5x + -3x2

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

Combine like terms: -13 + 13 = 0
13 + 5x + -3x2 = 0 + -5x + 5x + 3x2 + -3x2
13 + 5x + -3x2 = -5x + 5x + 3x2 + -3x2

Combine like terms: -5x + 5x = 0
13 + 5x + -3x2 = 0 + 3x2 + -3x2
13 + 5x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
13 + 5x + -3x2 = 0

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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