9x^2+36x+31=0

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Solution for 9x^2+36x+31=0 equation:


Simplifying
9x2 + 36x + 31 = 0

Reorder the terms:
31 + 36x + 9x2 = 0

Solving
31 + 36x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
3.444444444 + 4x + x2 = 0

Move the constant term to the right:

Add '-3.444444444' to each side of the equation.
3.444444444 + 4x + -3.444444444 + x2 = 0 + -3.444444444

Reorder the terms:
3.444444444 + -3.444444444 + 4x + x2 = 0 + -3.444444444

Combine like terms: 3.444444444 + -3.444444444 = 0.000000000
0.000000000 + 4x + x2 = 0 + -3.444444444
4x + x2 = 0 + -3.444444444

Combine like terms: 0 + -3.444444444 = -3.444444444
4x + x2 = -3.444444444

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = -3.444444444 + 4

Reorder the terms:
4 + 4x + x2 = -3.444444444 + 4

Combine like terms: -3.444444444 + 4 = 0.555555556
4 + 4x + x2 = 0.555555556

Factor a perfect square on the left side:
(x + 2)(x + 2) = 0.555555556

Calculate the square root of the right side: 0.745355993

Break this problem into two subproblems by setting 
(x + 2) equal to 0.745355993 and -0.745355993.

Subproblem 1

x + 2 = 0.745355993 Simplifying x + 2 = 0.745355993 Reorder the terms: 2 + x = 0.745355993 Solving 2 + x = 0.745355993 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = 0.745355993 + -2 Combine like terms: 2 + -2 = 0 0 + x = 0.745355993 + -2 x = 0.745355993 + -2 Combine like terms: 0.745355993 + -2 = -1.254644007 x = -1.254644007 Simplifying x = -1.254644007

Subproblem 2

x + 2 = -0.745355993 Simplifying x + 2 = -0.745355993 Reorder the terms: 2 + x = -0.745355993 Solving 2 + x = -0.745355993 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = -0.745355993 + -2 Combine like terms: 2 + -2 = 0 0 + x = -0.745355993 + -2 x = -0.745355993 + -2 Combine like terms: -0.745355993 + -2 = -2.745355993 x = -2.745355993 Simplifying x = -2.745355993

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.254644007, -2.745355993}

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