0=-24000+68x+-0.04x^2

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Solution for 0=-24000+68x+-0.04x^2 equation: 


Simplifying
0 = -24000 + 68x + -0.04x2

Solving
0 = -24000 + 68x + -0.04x2

Solving for variable 'x'.

Combine like terms: 0 + 24000 = 24000
24000 + -68x + 0.04x2 = -24000 + 68x + -0.04x2 + 24000 + -68x + 0.04x2

Reorder the terms:
24000 + -68x + 0.04x2 = -24000 + 24000 + 68x + -68x + -0.04x2 + 0.04x2

Combine like terms: -24000 + 24000 = 0
24000 + -68x + 0.04x2 = 0 + 68x + -68x + -0.04x2 + 0.04x2
24000 + -68x + 0.04x2 = 68x + -68x + -0.04x2 + 0.04x2

Combine like terms: 68x + -68x = 0
24000 + -68x + 0.04x2 = 0 + -0.04x2 + 0.04x2
24000 + -68x + 0.04x2 = -0.04x2 + 0.04x2

Combine like terms: -0.04x2 + 0.04x2 = 0.00
24000 + -68x + 0.04x2 = 0.00

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

Divide each side by '0.04'.
600000 + -1700x + x2 = 0

Move the constant term to the right:

Add '-600000' to each side of the equation.
600000 + -1700x + -600000 + x2 = 0 + -600000

Reorder the terms:
600000 + -600000 + -1700x + x2 = 0 + -600000

Combine like terms: 600000 + -600000 = 0
0 + -1700x + x2 = 0 + -600000
-1700x + x2 = 0 + -600000

Combine like terms: 0 + -600000 = -600000
-1700x + x2 = -600000

The x term is -1700x.  Take half its coefficient (-850).
Square it (722500) and add it to both sides.

Add '722500' to each side of the equation.
-1700x + 722500 + x2 = -600000 + 722500

Reorder the terms:
722500 + -1700x + x2 = -600000 + 722500

Combine like terms: -600000 + 722500 = 122500
722500 + -1700x + x2 = 122500

Factor a perfect square on the left side:
(x + -850)(x + -850) = 122500

Calculate the square root of the right side: 350

Break this problem into two subproblems by setting 
(x + -850) equal to 350 and -350.


Subproblem 1
x + -850 = 350

Simplifying
x + -850 = 350

Reorder the terms:
-850 + x = 350

Solving
-850 + x = 350

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '850' to each side of the equation.
-850 + 850 + x = 350 + 850

Combine like terms: -850 + 850 = 0
0 + x = 350 + 850
x = 350 + 850

Combine like terms: 350 + 850 = 1200
x = 1200

Simplifying
x = 1200


Subproblem 2
x + -850 = -350

Simplifying
x + -850 = -350

Reorder the terms:
-850 + x = -350

Solving
-850 + x = -350

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '850' to each side of the equation.
-850 + 850 + x = -350 + 850

Combine like terms: -850 + 850 = 0
0 + x = -350 + 850
x = -350 + 850

Combine like terms: -350 + 850 = 500
x = 500

Simplifying
x = 500


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1200, 500}

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