509=-0.5x^2+36x-103

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Solution for 509=-0.5x^2+36x-103 equation:


Simplifying
509 = -0.5x2 + 36x + -103

Reorder the terms:
509 = -103 + 36x + -0.5x2

Solving
509 = -103 + 36x + -0.5x2

Solving for variable 'x'.

Combine like terms: 509 + 103 = 612
612 + -36x + 0.5x2 = -103 + 36x + -0.5x2 + 103 + -36x + 0.5x2

Reorder the terms:
612 + -36x + 0.5x2 = -103 + 103 + 36x + -36x + -0.5x2 + 0.5x2

Combine like terms: -103 + 103 = 0
612 + -36x + 0.5x2 = 0 + 36x + -36x + -0.5x2 + 0.5x2
612 + -36x + 0.5x2 = 36x + -36x + -0.5x2 + 0.5x2

Combine like terms: 36x + -36x = 0
612 + -36x + 0.5x2 = 0 + -0.5x2 + 0.5x2
612 + -36x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
612 + -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'.
1224 + -72x + x2 = 0

Move the constant term to the right:

Add '-1224' to each side of the equation.
1224 + -72x + -1224 + x2 = 0 + -1224

Reorder the terms:
1224 + -1224 + -72x + x2 = 0 + -1224

Combine like terms: 1224 + -1224 = 0
0 + -72x + x2 = 0 + -1224
-72x + x2 = 0 + -1224

Combine like terms: 0 + -1224 = -1224
-72x + x2 = -1224

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 = -1224 + 1296

Reorder the terms:
1296 + -72x + x2 = -1224 + 1296

Combine like terms: -1224 + 1296 = 72
1296 + -72x + x2 = 72

Factor a perfect square on the left side:
(x + -36)(x + -36) = 72

Calculate the square root of the right side: 8.485281374

Break this problem into two subproblems by setting 
(x + -36) equal to 8.485281374 and -8.485281374.

Subproblem 1

x + -36 = 8.485281374 Simplifying x + -36 = 8.485281374 Reorder the terms: -36 + x = 8.485281374 Solving -36 + x = 8.485281374 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 = 8.485281374 + 36 Combine like terms: -36 + 36 = 0 0 + x = 8.485281374 + 36 x = 8.485281374 + 36 Combine like terms: 8.485281374 + 36 = 44.485281374 x = 44.485281374 Simplifying x = 44.485281374

Subproblem 2

x + -36 = -8.485281374 Simplifying x + -36 = -8.485281374 Reorder the terms: -36 + x = -8.485281374 Solving -36 + x = -8.485281374 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 = -8.485281374 + 36 Combine like terms: -36 + 36 = 0 0 + x = -8.485281374 + 36 x = -8.485281374 + 36 Combine like terms: -8.485281374 + 36 = 27.514718626 x = 27.514718626 Simplifying x = 27.514718626

Solution

The solution to the problem is based on the solutions from the subproblems. x = {44.485281374, 27.514718626}

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