2x^2+9x-108=0

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Solution for 2x^2+9x-108=0 equation:


Simplifying
2x2 + 9x + -108 = 0

Reorder the terms:
-108 + 9x + 2x2 = 0

Solving
-108 + 9x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-54 + 4.5x + x2 = 0

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + 4.5x + 54 + x2 = 0 + 54

Reorder the terms:
-54 + 54 + 4.5x + x2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + 4.5x + x2 = 0 + 54
4.5x + x2 = 0 + 54

Combine like terms: 0 + 54 = 54
4.5x + x2 = 54

The x term is 4.5x.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5x + 5.0625 + x2 = 54 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = 54 + 5.0625

Combine like terms: 54 + 5.0625 = 59.0625
5.0625 + 4.5x + x2 = 59.0625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 59.0625

Calculate the square root of the right side: 7.685213074

Break this problem into two subproblems by setting 
(x + 2.25) equal to 7.685213074 and -7.685213074.

Subproblem 1

x + 2.25 = 7.685213074 Simplifying x + 2.25 = 7.685213074 Reorder the terms: 2.25 + x = 7.685213074 Solving 2.25 + x = 7.685213074 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 7.685213074 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 7.685213074 + -2.25 x = 7.685213074 + -2.25 Combine like terms: 7.685213074 + -2.25 = 5.435213074 x = 5.435213074 Simplifying x = 5.435213074

Subproblem 2

x + 2.25 = -7.685213074 Simplifying x + 2.25 = -7.685213074 Reorder the terms: 2.25 + x = -7.685213074 Solving 2.25 + x = -7.685213074 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -7.685213074 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -7.685213074 + -2.25 x = -7.685213074 + -2.25 Combine like terms: -7.685213074 + -2.25 = -9.935213074 x = -9.935213074 Simplifying x = -9.935213074

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.435213074, -9.935213074}

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