5000=0.5x^2+0.5x

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Solution for 5000=0.5x^2+0.5x equation:


Simplifying
5000 = 0.5x2 + 0.5x

Reorder the terms:
5000 = 0.5x + 0.5x2

Solving
5000 = 0.5x + 0.5x2

Solving for variable 'x'.

Reorder the terms:
5000 + -0.5x + -0.5x2 = 0.5x + -0.5x + 0.5x2 + -0.5x2

Combine like terms: 0.5x + -0.5x = 0.0
5000 + -0.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2
5000 + -0.5x + -0.5x2 = 0.5x2 + -0.5x2

Combine like terms: 0.5x2 + -0.5x2 = 0.0
5000 + -0.5x + -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'.
-10000 + x + x2 = 0

Move the constant term to the right:

Add '10000' to each side of the equation.
-10000 + x + 10000 + x2 = 0 + 10000

Reorder the terms:
-10000 + 10000 + x + x2 = 0 + 10000

Combine like terms: -10000 + 10000 = 0
0 + x + x2 = 0 + 10000
x + x2 = 0 + 10000

Combine like terms: 0 + 10000 = 10000
x + x2 = 10000

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.
 + 0.25 + x2 = 10000 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 10000 + 0.25

Combine like terms: 10000 + 0.25 = 10000.25
1.25 + x2 = 10000.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 10000.25

Calculate the square root of the right side: 100.001249992

Break this problem into two subproblems by setting 
(x + 0.5) equal to 100.001249992 and -100.001249992.

Subproblem 1

x + 0.5 = 100.001249992 Simplifying x + 0.5 = 100.001249992 Reorder the terms: 0.5 + x = 100.001249992 Solving 0.5 + x = 100.001249992 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 = 100.001249992 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 100.001249992 + -0.5 x = 100.001249992 + -0.5 Combine like terms: 100.001249992 + -0.5 = 99.501249992 x = 99.501249992 Simplifying x = 99.501249992

Subproblem 2

x + 0.5 = -100.001249992 Simplifying x + 0.5 = -100.001249992 Reorder the terms: 0.5 + x = -100.001249992 Solving 0.5 + x = -100.001249992 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 = -100.001249992 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -100.001249992 + -0.5 x = -100.001249992 + -0.5 Combine like terms: -100.001249992 + -0.5 = -100.501249992 x = -100.501249992 Simplifying x = -100.501249992

Solution

The solution to the problem is based on the solutions from the subproblems. x = {99.501249992, -100.501249992}

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