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Simplifying [P(2000 + -3P)] + -1[5000 + 4(2000 + -3P)] = 0 [(2000 * P + -3P * P)] + -1[5000 + 4(2000 + -3P)] = 0 [(2000P + -3P2)] + -1[5000 + 4(2000 + -3P)] = 0 [2000P + -3P2] + -1[5000 + 4(2000 + -3P)] = 0 Remove brackets around [2000P + -3P2] 2000P + -3P2 + -1[5000 + 4(2000 + -3P)] = 0 2000P + -3P2 + -1[5000 + (2000 * 4 + -3P * 4)] = 0 2000P + -3P2 + -1[5000 + (8000 + -12P)] = 0 Combine like terms: 5000 + 8000 = 13000 2000P + -3P2 + -1[13000 + -12P] = 0 2000P + -3P2 + [13000 * -1 + -12P * -1] = 0 2000P + -3P2 + [-13000 + 12P] = 0 Reorder the terms: -13000 + 2000P + 12P + -3P2 = 0 Combine like terms: 2000P + 12P = 2012P -13000 + 2012P + -3P2 = 0 Solving -13000 + 2012P + -3P2 = 0 Solving for variable 'P'. Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 4333.333333 + -670.6666667P + P2 = 0 Move the constant term to the right: Add '-4333.333333' to each side of the equation. 4333.333333 + -670.6666667P + -4333.333333 + P2 = 0 + -4333.333333 Reorder the terms: 4333.333333 + -4333.333333 + -670.6666667P + P2 = 0 + -4333.333333 Combine like terms: 4333.333333 + -4333.333333 = 0.000000 0.000000 + -670.6666667P + P2 = 0 + -4333.333333 -670.6666667P + P2 = 0 + -4333.333333 Combine like terms: 0 + -4333.333333 = -4333.333333 -670.6666667P + P2 = -4333.333333 The P term is -670.6666667P. Take half its coefficient (-335.3333334). Square it (112448.4445) and add it to both sides. Add '112448.4445' to each side of the equation. -670.6666667P + 112448.4445 + P2 = -4333.333333 + 112448.4445 Reorder the terms: 112448.4445 + -670.6666667P + P2 = -4333.333333 + 112448.4445 Combine like terms: -4333.333333 + 112448.4445 = 108115.111167 112448.4445 + -670.6666667P + P2 = 108115.111167 Factor a perfect square on the left side: (P + -335.3333334)(P + -335.3333334) = 108115.111167 Calculate the square root of the right side: 328.808623924 Break this problem into two subproblems by setting (P + -335.3333334) equal to 328.808623924 and -328.808623924.Subproblem 1
P + -335.3333334 = 328.808623924 Simplifying P + -335.3333334 = 328.808623924 Reorder the terms: -335.3333334 + P = 328.808623924 Solving -335.3333334 + P = 328.808623924 Solving for variable 'P'. Move all terms containing P to the left, all other terms to the right. Add '335.3333334' to each side of the equation. -335.3333334 + 335.3333334 + P = 328.808623924 + 335.3333334 Combine like terms: -335.3333334 + 335.3333334 = 0.0000000 0.0000000 + P = 328.808623924 + 335.3333334 P = 328.808623924 + 335.3333334 Combine like terms: 328.808623924 + 335.3333334 = 664.141957324 P = 664.141957324 Simplifying P = 664.141957324Subproblem 2
P + -335.3333334 = -328.808623924 Simplifying P + -335.3333334 = -328.808623924 Reorder the terms: -335.3333334 + P = -328.808623924 Solving -335.3333334 + P = -328.808623924 Solving for variable 'P'. Move all terms containing P to the left, all other terms to the right. Add '335.3333334' to each side of the equation. -335.3333334 + 335.3333334 + P = -328.808623924 + 335.3333334 Combine like terms: -335.3333334 + 335.3333334 = 0.0000000 0.0000000 + P = -328.808623924 + 335.3333334 P = -328.808623924 + 335.3333334 Combine like terms: -328.808623924 + 335.3333334 = 6.524709476 P = 6.524709476 Simplifying P = 6.524709476Solution
The solution to the problem is based on the solutions from the subproblems. P = {664.141957324, 6.524709476}
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