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Simplifying 0 = -6p2 + 18000p + -1200000 Reorder the terms: 0 = -1200000 + 18000p + -6p2 Solving 0 = -1200000 + 18000p + -6p2 Solving for variable 'p'. Combine like terms: 0 + 1200000 = 1200000 1200000 + -18000p + 6p2 = -1200000 + 18000p + -6p2 + 1200000 + -18000p + 6p2 Reorder the terms: 1200000 + -18000p + 6p2 = -1200000 + 1200000 + 18000p + -18000p + -6p2 + 6p2 Combine like terms: -1200000 + 1200000 = 0 1200000 + -18000p + 6p2 = 0 + 18000p + -18000p + -6p2 + 6p2 1200000 + -18000p + 6p2 = 18000p + -18000p + -6p2 + 6p2 Combine like terms: 18000p + -18000p = 0 1200000 + -18000p + 6p2 = 0 + -6p2 + 6p2 1200000 + -18000p + 6p2 = -6p2 + 6p2 Combine like terms: -6p2 + 6p2 = 0 1200000 + -18000p + 6p2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(200000 + -3000p + p2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(200000 + -3000p + p2)' equal to zero and attempt to solve: Simplifying 200000 + -3000p + p2 = 0 Solving 200000 + -3000p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '-200000' to each side of the equation. 200000 + -3000p + -200000 + p2 = 0 + -200000 Reorder the terms: 200000 + -200000 + -3000p + p2 = 0 + -200000 Combine like terms: 200000 + -200000 = 0 0 + -3000p + p2 = 0 + -200000 -3000p + p2 = 0 + -200000 Combine like terms: 0 + -200000 = -200000 -3000p + p2 = -200000 The p term is -3000p. Take half its coefficient (-1500). Square it (2250000) and add it to both sides. Add '2250000' to each side of the equation. -3000p + 2250000 + p2 = -200000 + 2250000 Reorder the terms: 2250000 + -3000p + p2 = -200000 + 2250000 Combine like terms: -200000 + 2250000 = 2050000 2250000 + -3000p + p2 = 2050000 Factor a perfect square on the left side: (p + -1500)(p + -1500) = 2050000 Calculate the square root of the right side: 1431.782106328 Break this problem into two subproblems by setting (p + -1500) equal to 1431.782106328 and -1431.782106328.Subproblem 1
p + -1500 = 1431.782106328 Simplifying p + -1500 = 1431.782106328 Reorder the terms: -1500 + p = 1431.782106328 Solving -1500 + p = 1431.782106328 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1500' to each side of the equation. -1500 + 1500 + p = 1431.782106328 + 1500 Combine like terms: -1500 + 1500 = 0 0 + p = 1431.782106328 + 1500 p = 1431.782106328 + 1500 Combine like terms: 1431.782106328 + 1500 = 2931.782106328 p = 2931.782106328 Simplifying p = 2931.782106328Subproblem 2
p + -1500 = -1431.782106328 Simplifying p + -1500 = -1431.782106328 Reorder the terms: -1500 + p = -1431.782106328 Solving -1500 + p = -1431.782106328 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1500' to each side of the equation. -1500 + 1500 + p = -1431.782106328 + 1500 Combine like terms: -1500 + 1500 = 0 0 + p = -1431.782106328 + 1500 p = -1431.782106328 + 1500 Combine like terms: -1431.782106328 + 1500 = 68.217893672 p = 68.217893672 Simplifying p = 68.217893672Solution
The solution to the problem is based on the solutions from the subproblems. p = {2931.782106328, 68.217893672}Solution
p = {2931.782106328, 68.217893672}
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