116412=505x-0.5x^2

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Solution for 116412=505x-0.5x^2 equation:


Simplifying
116412 = 505x + -0.5x2

Solving
116412 = 505x + -0.5x2

Solving for variable 'x'.

Reorder the terms:
116412 + -505x + 0.5x2 = 505x + -505x + -0.5x2 + 0.5x2

Combine like terms: 505x + -505x = 0
116412 + -505x + 0.5x2 = 0 + -0.5x2 + 0.5x2
116412 + -505x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
116412 + -505x + 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'.
232824 + -1010x + x2 = 0

Move the constant term to the right:

Add '-232824' to each side of the equation.
232824 + -1010x + -232824 + x2 = 0 + -232824

Reorder the terms:
232824 + -232824 + -1010x + x2 = 0 + -232824

Combine like terms: 232824 + -232824 = 0
0 + -1010x + x2 = 0 + -232824
-1010x + x2 = 0 + -232824

Combine like terms: 0 + -232824 = -232824
-1010x + x2 = -232824

The x term is -1010x.  Take half its coefficient (-505).
Square it (255025) and add it to both sides.

Add '255025' to each side of the equation.
-1010x + 255025 + x2 = -232824 + 255025

Reorder the terms:
255025 + -1010x + x2 = -232824 + 255025

Combine like terms: -232824 + 255025 = 22201
255025 + -1010x + x2 = 22201

Factor a perfect square on the left side:
(x + -505)(x + -505) = 22201

Calculate the square root of the right side: 149

Break this problem into two subproblems by setting 
(x + -505) equal to 149 and -149.

Subproblem 1

x + -505 = 149 Simplifying x + -505 = 149 Reorder the terms: -505 + x = 149 Solving -505 + x = 149 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '505' to each side of the equation. -505 + 505 + x = 149 + 505 Combine like terms: -505 + 505 = 0 0 + x = 149 + 505 x = 149 + 505 Combine like terms: 149 + 505 = 654 x = 654 Simplifying x = 654

Subproblem 2

x + -505 = -149 Simplifying x + -505 = -149 Reorder the terms: -505 + x = -149 Solving -505 + x = -149 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '505' to each side of the equation. -505 + 505 + x = -149 + 505 Combine like terms: -505 + 505 = 0 0 + x = -149 + 505 x = -149 + 505 Combine like terms: -149 + 505 = 356 x = 356 Simplifying x = 356

Solution

The solution to the problem is based on the solutions from the subproblems. x = {654, 356}

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