31=6x^2+33x

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Solution for 31=6x^2+33x equation:


Simplifying
31 = 6x2 + 33x

Reorder the terms:
31 = 33x + 6x2

Solving
31 = 33x + 6x2

Solving for variable 'x'.

Reorder the terms:
31 + -33x + -6x2 = 33x + -33x + 6x2 + -6x2

Combine like terms: 33x + -33x = 0
31 + -33x + -6x2 = 0 + 6x2 + -6x2
31 + -33x + -6x2 = 6x2 + -6x2

Combine like terms: 6x2 + -6x2 = 0
31 + -33x + -6x2 = 0

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
-5.166666667 + 5.5x + x2 = 0

Move the constant term to the right:

Add '5.166666667' to each side of the equation.
-5.166666667 + 5.5x + 5.166666667 + x2 = 0 + 5.166666667

Reorder the terms:
-5.166666667 + 5.166666667 + 5.5x + x2 = 0 + 5.166666667

Combine like terms: -5.166666667 + 5.166666667 = 0.000000000
0.000000000 + 5.5x + x2 = 0 + 5.166666667
5.5x + x2 = 0 + 5.166666667

Combine like terms: 0 + 5.166666667 = 5.166666667
5.5x + x2 = 5.166666667

The x term is 5.5x.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5x + 7.5625 + x2 = 5.166666667 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = 5.166666667 + 7.5625

Combine like terms: 5.166666667 + 7.5625 = 12.729166667
7.5625 + 5.5x + x2 = 12.729166667

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 12.729166667

Calculate the square root of the right side: 3.567795771

Break this problem into two subproblems by setting 
(x + 2.75) equal to 3.567795771 and -3.567795771.

Subproblem 1

x + 2.75 = 3.567795771 Simplifying x + 2.75 = 3.567795771 Reorder the terms: 2.75 + x = 3.567795771 Solving 2.75 + x = 3.567795771 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = 3.567795771 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = 3.567795771 + -2.75 x = 3.567795771 + -2.75 Combine like terms: 3.567795771 + -2.75 = 0.817795771 x = 0.817795771 Simplifying x = 0.817795771

Subproblem 2

x + 2.75 = -3.567795771 Simplifying x + 2.75 = -3.567795771 Reorder the terms: 2.75 + x = -3.567795771 Solving 2.75 + x = -3.567795771 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = -3.567795771 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = -3.567795771 + -2.75 x = -3.567795771 + -2.75 Combine like terms: -3.567795771 + -2.75 = -6.317795771 x = -6.317795771 Simplifying x = -6.317795771

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.817795771, -6.317795771}

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