(100x-0.5x^2)-(60x+300)=300

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Solution for (100x-0.5x^2)-(60x+300)=300 equation:


Simplifying
(100x + -0.5x2) + -1(60x + 300) = 300

Remove parenthesis around (100x + -0.5x2)
100x + -0.5x2 + -1(60x + 300) = 300

Reorder the terms:
100x + -0.5x2 + -1(300 + 60x) = 300
100x + -0.5x2 + (300 * -1 + 60x * -1) = 300
100x + -0.5x2 + (-300 + -60x) = 300

Reorder the terms:
-300 + 100x + -60x + -0.5x2 = 300

Combine like terms: 100x + -60x = 40x
-300 + 40x + -0.5x2 = 300

Solving
-300 + 40x + -0.5x2 = 300

Solving for variable 'x'.

Reorder the terms:
-300 + -300 + 40x + -0.5x2 = 300 + -300

Combine like terms: -300 + -300 = -600
-600 + 40x + -0.5x2 = 300 + -300

Combine like terms: 300 + -300 = 0
-600 + 40x + -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'.
1200 + -80x + x2 = 0

Move the constant term to the right:

Add '-1200' to each side of the equation.
1200 + -80x + -1200 + x2 = 0 + -1200

Reorder the terms:
1200 + -1200 + -80x + x2 = 0 + -1200

Combine like terms: 1200 + -1200 = 0
0 + -80x + x2 = 0 + -1200
-80x + x2 = 0 + -1200

Combine like terms: 0 + -1200 = -1200
-80x + x2 = -1200

The x term is -80x.  Take half its coefficient (-40).
Square it (1600) and add it to both sides.

Add '1600' to each side of the equation.
-80x + 1600 + x2 = -1200 + 1600

Reorder the terms:
1600 + -80x + x2 = -1200 + 1600

Combine like terms: -1200 + 1600 = 400
1600 + -80x + x2 = 400

Factor a perfect square on the left side:
(x + -40)(x + -40) = 400

Calculate the square root of the right side: 20

Break this problem into two subproblems by setting 
(x + -40) equal to 20 and -20.

Subproblem 1

x + -40 = 20 Simplifying x + -40 = 20 Reorder the terms: -40 + x = 20 Solving -40 + x = 20 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '40' to each side of the equation. -40 + 40 + x = 20 + 40 Combine like terms: -40 + 40 = 0 0 + x = 20 + 40 x = 20 + 40 Combine like terms: 20 + 40 = 60 x = 60 Simplifying x = 60

Subproblem 2

x + -40 = -20 Simplifying x + -40 = -20 Reorder the terms: -40 + x = -20 Solving -40 + x = -20 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '40' to each side of the equation. -40 + 40 + x = -20 + 40 Combine like terms: -40 + 40 = 0 0 + x = -20 + 40 x = -20 + 40 Combine like terms: -20 + 40 = 20 x = 20 Simplifying x = 20

Solution

The solution to the problem is based on the solutions from the subproblems. x = {60, 20}

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