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Simplifying 500 + p + -3p2 = 0 Solving 500 + p + -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'. -166.6666667 + -0.3333333333p + p2 = 0 Move the constant term to the right: Add '166.6666667' to each side of the equation. -166.6666667 + -0.3333333333p + 166.6666667 + p2 = 0 + 166.6666667 Reorder the terms: -166.6666667 + 166.6666667 + -0.3333333333p + p2 = 0 + 166.6666667 Combine like terms: -166.6666667 + 166.6666667 = 0.0000000 0.0000000 + -0.3333333333p + p2 = 0 + 166.6666667 -0.3333333333p + p2 = 0 + 166.6666667 Combine like terms: 0 + 166.6666667 = 166.6666667 -0.3333333333p + p2 = 166.6666667 The p term is p. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.3333333333p + 0.25 + p2 = 166.6666667 + 0.25 Reorder the terms: 0.25 + -0.3333333333p + p2 = 166.6666667 + 0.25 Combine like terms: 166.6666667 + 0.25 = 166.9166667 0.25 + -0.3333333333p + p2 = 166.9166667 Factor a perfect square on the left side: (p + 0.5)(p + 0.5) = 166.9166667 Calculate the square root of the right side: 12.919623319 Break this problem into two subproblems by setting (p + 0.5) equal to 12.919623319 and -12.919623319.Subproblem 1
p + 0.5 = 12.919623319 Simplifying p + 0.5 = 12.919623319 Reorder the terms: 0.5 + p = 12.919623319 Solving 0.5 + p = 12.919623319 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + p = 12.919623319 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = 12.919623319 + -0.5 p = 12.919623319 + -0.5 Combine like terms: 12.919623319 + -0.5 = 12.419623319 p = 12.419623319 Simplifying p = 12.419623319Subproblem 2
p + 0.5 = -12.919623319 Simplifying p + 0.5 = -12.919623319 Reorder the terms: 0.5 + p = -12.919623319 Solving 0.5 + p = -12.919623319 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + p = -12.919623319 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = -12.919623319 + -0.5 p = -12.919623319 + -0.5 Combine like terms: -12.919623319 + -0.5 = -13.419623319 p = -13.419623319 Simplifying p = -13.419623319Solution
The solution to the problem is based on the solutions from the subproblems. p = {12.419623319, -13.419623319}
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