3x^2-18x-31=0

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Solution for 3x^2-18x-31=0 equation:


Simplifying
3x2 + -18x + -31 = 0

Reorder the terms:
-31 + -18x + 3x2 = 0

Solving
-31 + -18x + 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'.
-10.33333333 + -6x + x2 = 0

Move the constant term to the right:

Add '10.33333333' to each side of the equation.
-10.33333333 + -6x + 10.33333333 + x2 = 0 + 10.33333333

Reorder the terms:
-10.33333333 + 10.33333333 + -6x + x2 = 0 + 10.33333333

Combine like terms: -10.33333333 + 10.33333333 = 0.00000000
0.00000000 + -6x + x2 = 0 + 10.33333333
-6x + x2 = 0 + 10.33333333

Combine like terms: 0 + 10.33333333 = 10.33333333
-6x + x2 = 10.33333333

The x term is -6x.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x + 9 + x2 = 10.33333333 + 9

Reorder the terms:
9 + -6x + x2 = 10.33333333 + 9

Combine like terms: 10.33333333 + 9 = 19.33333333
9 + -6x + x2 = 19.33333333

Factor a perfect square on the left side:
(x + -3)(x + -3) = 19.33333333

Calculate the square root of the right side: 4.396968652

Break this problem into two subproblems by setting 
(x + -3) equal to 4.396968652 and -4.396968652.

Subproblem 1

x + -3 = 4.396968652 Simplifying x + -3 = 4.396968652 Reorder the terms: -3 + x = 4.396968652 Solving -3 + x = 4.396968652 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = 4.396968652 + 3 Combine like terms: -3 + 3 = 0 0 + x = 4.396968652 + 3 x = 4.396968652 + 3 Combine like terms: 4.396968652 + 3 = 7.396968652 x = 7.396968652 Simplifying x = 7.396968652

Subproblem 2

x + -3 = -4.396968652 Simplifying x + -3 = -4.396968652 Reorder the terms: -3 + x = -4.396968652 Solving -3 + x = -4.396968652 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = -4.396968652 + 3 Combine like terms: -3 + 3 = 0 0 + x = -4.396968652 + 3 x = -4.396968652 + 3 Combine like terms: -4.396968652 + 3 = -1.396968652 x = -1.396968652 Simplifying x = -1.396968652

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.396968652, -1.396968652}

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