0=-.5x^2+36x-109

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Solution for 0=-.5x^2+36x-109 equation:


Simplifying
0 = -0.5x2 + 36x + -109

Reorder the terms:
0 = -109 + 36x + -0.5x2

Solving
0 = -109 + 36x + -0.5x2

Solving for variable 'x'.

Combine like terms: 0 + 109 = 109
109 + -36x + 0.5x2 = -109 + 36x + -0.5x2 + 109 + -36x + 0.5x2

Reorder the terms:
109 + -36x + 0.5x2 = -109 + 109 + 36x + -36x + -0.5x2 + 0.5x2

Combine like terms: -109 + 109 = 0
109 + -36x + 0.5x2 = 0 + 36x + -36x + -0.5x2 + 0.5x2
109 + -36x + 0.5x2 = 36x + -36x + -0.5x2 + 0.5x2

Combine like terms: 36x + -36x = 0
109 + -36x + 0.5x2 = 0 + -0.5x2 + 0.5x2
109 + -36x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
109 + -36x + 0.5x2 = 0.0

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

Divide each side by '0.5'.
218 + -72x + x2 = 0

Move the constant term to the right:

Add '-218' to each side of the equation.
218 + -72x + -218 + x2 = 0 + -218

Reorder the terms:
218 + -218 + -72x + x2 = 0 + -218

Combine like terms: 218 + -218 = 0
0 + -72x + x2 = 0 + -218
-72x + x2 = 0 + -218

Combine like terms: 0 + -218 = -218
-72x + x2 = -218

The x term is -72x.  Take half its coefficient (-36).
Square it (1296) and add it to both sides.

Add '1296' to each side of the equation.
-72x + 1296 + x2 = -218 + 1296

Reorder the terms:
1296 + -72x + x2 = -218 + 1296

Combine like terms: -218 + 1296 = 1078
1296 + -72x + x2 = 1078

Factor a perfect square on the left side:
(x + -36)(x + -36) = 1078

Calculate the square root of the right side: 32.832910319

Break this problem into two subproblems by setting 
(x + -36) equal to 32.832910319 and -32.832910319.

Subproblem 1

x + -36 = 32.832910319 Simplifying x + -36 = 32.832910319 Reorder the terms: -36 + x = 32.832910319 Solving -36 + x = 32.832910319 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '36' to each side of the equation. -36 + 36 + x = 32.832910319 + 36 Combine like terms: -36 + 36 = 0 0 + x = 32.832910319 + 36 x = 32.832910319 + 36 Combine like terms: 32.832910319 + 36 = 68.832910319 x = 68.832910319 Simplifying x = 68.832910319

Subproblem 2

x + -36 = -32.832910319 Simplifying x + -36 = -32.832910319 Reorder the terms: -36 + x = -32.832910319 Solving -36 + x = -32.832910319 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '36' to each side of the equation. -36 + 36 + x = -32.832910319 + 36 Combine like terms: -36 + 36 = 0 0 + x = -32.832910319 + 36 x = -32.832910319 + 36 Combine like terms: -32.832910319 + 36 = 3.167089681 x = 3.167089681 Simplifying x = 3.167089681

Solution

The solution to the problem is based on the solutions from the subproblems. x = {68.832910319, 3.167089681}

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