0=3x^2-2x-31

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Solution for 0=3x^2-2x-31 equation:


Simplifying
0 = 3x2 + -2x + -31

Reorder the terms:
0 = -31 + -2x + 3x2

Solving
0 = -31 + -2x + 3x2

Solving for variable 'x'.

Combine like terms: 0 + 31 = 31
31 + 2x + -3x2 = -31 + -2x + 3x2 + 31 + 2x + -3x2

Reorder the terms:
31 + 2x + -3x2 = -31 + 31 + -2x + 2x + 3x2 + -3x2

Combine like terms: -31 + 31 = 0
31 + 2x + -3x2 = 0 + -2x + 2x + 3x2 + -3x2
31 + 2x + -3x2 = -2x + 2x + 3x2 + -3x2

Combine like terms: -2x + 2x = 0
31 + 2x + -3x2 = 0 + 3x2 + -3x2
31 + 2x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
31 + 2x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-10.33333333 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '10.33333333' to each side of the equation.
-10.33333333 + -0.6666666667x + 10.33333333 + x2 = 0 + 10.33333333

Reorder the terms:
-10.33333333 + 10.33333333 + -0.6666666667x + x2 = 0 + 10.33333333

Combine like terms: -10.33333333 + 10.33333333 = 0.00000000
0.00000000 + -0.6666666667x + x2 = 0 + 10.33333333
-0.6666666667x + x2 = 0 + 10.33333333

Combine like terms: 0 + 10.33333333 = 10.33333333
-0.6666666667x + x2 = 10.33333333

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 10.33333333 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 10.33333333 + 0.1111111112

Combine like terms: 10.33333333 + 0.1111111112 = 10.4444444412
0.1111111112 + -0.6666666667x + x2 = 10.4444444412

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 10.4444444412

Calculate the square root of the right side: 3.231786571

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 3.231786571 and -3.231786571.

Subproblem 1

x + -0.3333333334 = 3.231786571 Simplifying x + -0.3333333334 = 3.231786571 Reorder the terms: -0.3333333334 + x = 3.231786571 Solving -0.3333333334 + x = 3.231786571 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 3.231786571 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 3.231786571 + 0.3333333334 x = 3.231786571 + 0.3333333334 Combine like terms: 3.231786571 + 0.3333333334 = 3.5651199044 x = 3.5651199044 Simplifying x = 3.5651199044

Subproblem 2

x + -0.3333333334 = -3.231786571 Simplifying x + -0.3333333334 = -3.231786571 Reorder the terms: -0.3333333334 + x = -3.231786571 Solving -0.3333333334 + x = -3.231786571 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -3.231786571 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -3.231786571 + 0.3333333334 x = -3.231786571 + 0.3333333334 Combine like terms: -3.231786571 + 0.3333333334 = -2.8984532376 x = -2.8984532376 Simplifying x = -2.8984532376

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.5651199044, -2.8984532376}

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