x+0.5x^2=250

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Solution for x+0.5x^2=250 equation:


Simplifying
x + 0.5x2 = 250

Solving
x + 0.5x2 = 250

Solving for variable 'x'.

Reorder the terms:
-250 + x + 0.5x2 = 250 + -250

Combine like terms: 250 + -250 = 0
-250 + x + 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'.
-500 + 2x + x2 = 0

Move the constant term to the right:

Add '500' to each side of the equation.
-500 + 2x + 500 + x2 = 0 + 500

Reorder the terms:
-500 + 500 + 2x + x2 = 0 + 500

Combine like terms: -500 + 500 = 0
0 + 2x + x2 = 0 + 500
2x + x2 = 0 + 500

Combine like terms: 0 + 500 = 500
2x + x2 = 500

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
2x + 0.25 + x2 = 500 + 0.25

Reorder the terms:
0.25 + 2x + x2 = 500 + 0.25

Combine like terms: 500 + 0.25 = 500.25
0.25 + 2x + x2 = 500.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 500.25

Calculate the square root of the right side: 22.366269246

Break this problem into two subproblems by setting 
(x + 0.5) equal to 22.366269246 and -22.366269246.

Subproblem 1

x + 0.5 = 22.366269246 Simplifying x + 0.5 = 22.366269246 Reorder the terms: 0.5 + x = 22.366269246 Solving 0.5 + x = 22.366269246 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 22.366269246 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 22.366269246 + -0.5 x = 22.366269246 + -0.5 Combine like terms: 22.366269246 + -0.5 = 21.866269246 x = 21.866269246 Simplifying x = 21.866269246

Subproblem 2

x + 0.5 = -22.366269246 Simplifying x + 0.5 = -22.366269246 Reorder the terms: 0.5 + x = -22.366269246 Solving 0.5 + x = -22.366269246 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -22.366269246 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -22.366269246 + -0.5 x = -22.366269246 + -0.5 Combine like terms: -22.366269246 + -0.5 = -22.866269246 x = -22.866269246 Simplifying x = -22.866269246

Solution

The solution to the problem is based on the solutions from the subproblems. x = {21.866269246, -22.866269246}

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