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Simplifying 0 = -500x2 + 10000x + 100000 Reorder the terms: 0 = 100000 + 10000x + -500x2 Solving 0 = 100000 + 10000x + -500x2 Solving for variable 'x'. Combine like terms: 0 + -100000 = -100000 -100000 + -10000x + 500x2 = 100000 + 10000x + -500x2 + -100000 + -10000x + 500x2 Reorder the terms: -100000 + -10000x + 500x2 = 100000 + -100000 + 10000x + -10000x + -500x2 + 500x2 Combine like terms: 100000 + -100000 = 0 -100000 + -10000x + 500x2 = 0 + 10000x + -10000x + -500x2 + 500x2 -100000 + -10000x + 500x2 = 10000x + -10000x + -500x2 + 500x2 Combine like terms: 10000x + -10000x = 0 -100000 + -10000x + 500x2 = 0 + -500x2 + 500x2 -100000 + -10000x + 500x2 = -500x2 + 500x2 Combine like terms: -500x2 + 500x2 = 0 -100000 + -10000x + 500x2 = 0 Factor out the Greatest Common Factor (GCF), '500'. 500(-200 + -20x + x2) = 0 Ignore the factor 500.Subproblem 1
Set the factor '(-200 + -20x + x2)' equal to zero and attempt to solve: Simplifying -200 + -20x + x2 = 0 Solving -200 + -20x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '200' to each side of the equation. -200 + -20x + 200 + x2 = 0 + 200 Reorder the terms: -200 + 200 + -20x + x2 = 0 + 200 Combine like terms: -200 + 200 = 0 0 + -20x + x2 = 0 + 200 -20x + x2 = 0 + 200 Combine like terms: 0 + 200 = 200 -20x + x2 = 200 The x term is -20x. Take half its coefficient (-10). Square it (100) and add it to both sides. Add '100' to each side of the equation. -20x + 100 + x2 = 200 + 100 Reorder the terms: 100 + -20x + x2 = 200 + 100 Combine like terms: 200 + 100 = 300 100 + -20x + x2 = 300 Factor a perfect square on the left side: (x + -10)(x + -10) = 300 Calculate the square root of the right side: 17.320508076 Break this problem into two subproblems by setting (x + -10) equal to 17.320508076 and -17.320508076.Subproblem 1
x + -10 = 17.320508076 Simplifying x + -10 = 17.320508076 Reorder the terms: -10 + x = 17.320508076 Solving -10 + x = 17.320508076 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x = 17.320508076 + 10 Combine like terms: -10 + 10 = 0 0 + x = 17.320508076 + 10 x = 17.320508076 + 10 Combine like terms: 17.320508076 + 10 = 27.320508076 x = 27.320508076 Simplifying x = 27.320508076Subproblem 2
x + -10 = -17.320508076 Simplifying x + -10 = -17.320508076 Reorder the terms: -10 + x = -17.320508076 Solving -10 + x = -17.320508076 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x = -17.320508076 + 10 Combine like terms: -10 + 10 = 0 0 + x = -17.320508076 + 10 x = -17.320508076 + 10 Combine like terms: -17.320508076 + 10 = -7.320508076 x = -7.320508076 Simplifying x = -7.320508076Solution
The solution to the problem is based on the solutions from the subproblems. x = {27.320508076, -7.320508076}Solution
x = {27.320508076, -7.320508076}
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