489x-0.5x^2=110980

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Solution for 489x-0.5x^2=110980 equation:


Simplifying
489x + -0.5x2 = 110980

Solving
489x + -0.5x2 = 110980

Solving for variable 'x'.

Reorder the terms:
-110980 + 489x + -0.5x2 = 110980 + -110980

Combine like terms: 110980 + -110980 = 0
-110980 + 489x + -0.5x2 = 0

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

Divide each side by '-0.5'.
221960 + -978x + x2 = 0

Move the constant term to the right:

Add '-221960' to each side of the equation.
221960 + -978x + -221960 + x2 = 0 + -221960

Reorder the terms:
221960 + -221960 + -978x + x2 = 0 + -221960

Combine like terms: 221960 + -221960 = 0
0 + -978x + x2 = 0 + -221960
-978x + x2 = 0 + -221960

Combine like terms: 0 + -221960 = -221960
-978x + x2 = -221960

The x term is -978x.  Take half its coefficient (-489).
Square it (239121) and add it to both sides.

Add '239121' to each side of the equation.
-978x + 239121 + x2 = -221960 + 239121

Reorder the terms:
239121 + -978x + x2 = -221960 + 239121

Combine like terms: -221960 + 239121 = 17161
239121 + -978x + x2 = 17161

Factor a perfect square on the left side:
(x + -489)(x + -489) = 17161

Calculate the square root of the right side: 131

Break this problem into two subproblems by setting 
(x + -489) equal to 131 and -131.

Subproblem 1

x + -489 = 131 Simplifying x + -489 = 131 Reorder the terms: -489 + x = 131 Solving -489 + x = 131 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '489' to each side of the equation. -489 + 489 + x = 131 + 489 Combine like terms: -489 + 489 = 0 0 + x = 131 + 489 x = 131 + 489 Combine like terms: 131 + 489 = 620 x = 620 Simplifying x = 620

Subproblem 2

x + -489 = -131 Simplifying x + -489 = -131 Reorder the terms: -489 + x = -131 Solving -489 + x = -131 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '489' to each side of the equation. -489 + 489 + x = -131 + 489 Combine like terms: -489 + 489 = 0 0 + x = -131 + 489 x = -131 + 489 Combine like terms: -131 + 489 = 358 x = 358 Simplifying x = 358

Solution

The solution to the problem is based on the solutions from the subproblems. x = {620, 358}

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