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Simplifying 15(p) = 2000 + -4p + -1p2 Solving 15p = 2000 + -4p + -1p2 Solving for variable 'p'. Reorder the terms: -2000 + 15p + 4p + p2 = 2000 + -4p + -1p2 + -2000 + 4p + p2 Combine like terms: 15p + 4p = 19p -2000 + 19p + p2 = 2000 + -4p + -1p2 + -2000 + 4p + p2 Reorder the terms: -2000 + 19p + p2 = 2000 + -2000 + -4p + 4p + -1p2 + p2 Combine like terms: 2000 + -2000 = 0 -2000 + 19p + p2 = 0 + -4p + 4p + -1p2 + p2 -2000 + 19p + p2 = -4p + 4p + -1p2 + p2 Combine like terms: -4p + 4p = 0 -2000 + 19p + p2 = 0 + -1p2 + p2 -2000 + 19p + p2 = -1p2 + p2 Combine like terms: -1p2 + p2 = 0 -2000 + 19p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '2000' to each side of the equation. -2000 + 19p + 2000 + p2 = 0 + 2000 Reorder the terms: -2000 + 2000 + 19p + p2 = 0 + 2000 Combine like terms: -2000 + 2000 = 0 0 + 19p + p2 = 0 + 2000 19p + p2 = 0 + 2000 Combine like terms: 0 + 2000 = 2000 19p + p2 = 2000 The p term is 19p. Take half its coefficient (9.5). Square it (90.25) and add it to both sides. Add '90.25' to each side of the equation. 19p + 90.25 + p2 = 2000 + 90.25 Reorder the terms: 90.25 + 19p + p2 = 2000 + 90.25 Combine like terms: 2000 + 90.25 = 2090.25 90.25 + 19p + p2 = 2090.25 Factor a perfect square on the left side: (p + 9.5)(p + 9.5) = 2090.25 Calculate the square root of the right side: 45.719251962 Break this problem into two subproblems by setting (p + 9.5) equal to 45.719251962 and -45.719251962.Subproblem 1
p + 9.5 = 45.719251962 Simplifying p + 9.5 = 45.719251962 Reorder the terms: 9.5 + p = 45.719251962 Solving 9.5 + p = 45.719251962 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + p = 45.719251962 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + p = 45.719251962 + -9.5 p = 45.719251962 + -9.5 Combine like terms: 45.719251962 + -9.5 = 36.219251962 p = 36.219251962 Simplifying p = 36.219251962Subproblem 2
p + 9.5 = -45.719251962 Simplifying p + 9.5 = -45.719251962 Reorder the terms: 9.5 + p = -45.719251962 Solving 9.5 + p = -45.719251962 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + p = -45.719251962 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + p = -45.719251962 + -9.5 p = -45.719251962 + -9.5 Combine like terms: -45.719251962 + -9.5 = -55.219251962 p = -55.219251962 Simplifying p = -55.219251962Solution
The solution to the problem is based on the solutions from the subproblems. p = {36.219251962, -55.219251962}
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