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Simplifying 27 = -3p2 + 162p + -275 Reorder the terms: 27 = -275 + 162p + -3p2 Solving 27 = -275 + 162p + -3p2 Solving for variable 'p'. Combine like terms: 27 + 275 = 302 302 + -162p + 3p2 = -275 + 162p + -3p2 + 275 + -162p + 3p2 Reorder the terms: 302 + -162p + 3p2 = -275 + 275 + 162p + -162p + -3p2 + 3p2 Combine like terms: -275 + 275 = 0 302 + -162p + 3p2 = 0 + 162p + -162p + -3p2 + 3p2 302 + -162p + 3p2 = 162p + -162p + -3p2 + 3p2 Combine like terms: 162p + -162p = 0 302 + -162p + 3p2 = 0 + -3p2 + 3p2 302 + -162p + 3p2 = -3p2 + 3p2 Combine like terms: -3p2 + 3p2 = 0 302 + -162p + 3p2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 100.6666667 + -54p + p2 = 0 Move the constant term to the right: Add '-100.6666667' to each side of the equation. 100.6666667 + -54p + -100.6666667 + p2 = 0 + -100.6666667 Reorder the terms: 100.6666667 + -100.6666667 + -54p + p2 = 0 + -100.6666667 Combine like terms: 100.6666667 + -100.6666667 = 0.0000000 0.0000000 + -54p + p2 = 0 + -100.6666667 -54p + p2 = 0 + -100.6666667 Combine like terms: 0 + -100.6666667 = -100.6666667 -54p + p2 = -100.6666667 The p term is -54p. Take half its coefficient (-27). Square it (729) and add it to both sides. Add '729' to each side of the equation. -54p + 729 + p2 = -100.6666667 + 729 Reorder the terms: 729 + -54p + p2 = -100.6666667 + 729 Combine like terms: -100.6666667 + 729 = 628.3333333 729 + -54p + p2 = 628.3333333 Factor a perfect square on the left side: (p + -27)(p + -27) = 628.3333333 Calculate the square root of the right side: 25.066578013 Break this problem into two subproblems by setting (p + -27) equal to 25.066578013 and -25.066578013.Subproblem 1
p + -27 = 25.066578013 Simplifying p + -27 = 25.066578013 Reorder the terms: -27 + p = 25.066578013 Solving -27 + p = 25.066578013 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '27' to each side of the equation. -27 + 27 + p = 25.066578013 + 27 Combine like terms: -27 + 27 = 0 0 + p = 25.066578013 + 27 p = 25.066578013 + 27 Combine like terms: 25.066578013 + 27 = 52.066578013 p = 52.066578013 Simplifying p = 52.066578013Subproblem 2
p + -27 = -25.066578013 Simplifying p + -27 = -25.066578013 Reorder the terms: -27 + p = -25.066578013 Solving -27 + p = -25.066578013 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '27' to each side of the equation. -27 + 27 + p = -25.066578013 + 27 Combine like terms: -27 + 27 = 0 0 + p = -25.066578013 + 27 p = -25.066578013 + 27 Combine like terms: -25.066578013 + 27 = 1.933421987 p = 1.933421987 Simplifying p = 1.933421987Solution
The solution to the problem is based on the solutions from the subproblems. p = {52.066578013, 1.933421987}
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