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Simplifying 3(p2 + -18p + 81) = 81 Reorder the terms: 3(81 + -18p + p2) = 81 (81 * 3 + -18p * 3 + p2 * 3) = 81 (243 + -54p + 3p2) = 81 Solving 243 + -54p + 3p2 = 81 Solving for variable 'p'. Reorder the terms: 243 + -81 + -54p + 3p2 = 81 + -81 Combine like terms: 243 + -81 = 162 162 + -54p + 3p2 = 81 + -81 Combine like terms: 81 + -81 = 0 162 + -54p + 3p2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(54 + -18p + p2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(54 + -18p + p2)' equal to zero and attempt to solve: Simplifying 54 + -18p + p2 = 0 Solving 54 + -18p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '-54' to each side of the equation. 54 + -18p + -54 + p2 = 0 + -54 Reorder the terms: 54 + -54 + -18p + p2 = 0 + -54 Combine like terms: 54 + -54 = 0 0 + -18p + p2 = 0 + -54 -18p + p2 = 0 + -54 Combine like terms: 0 + -54 = -54 -18p + p2 = -54 The p term is -18p. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18p + 81 + p2 = -54 + 81 Reorder the terms: 81 + -18p + p2 = -54 + 81 Combine like terms: -54 + 81 = 27 81 + -18p + p2 = 27 Factor a perfect square on the left side: (p + -9)(p + -9) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (p + -9) equal to 5.196152423 and -5.196152423.Subproblem 1
p + -9 = 5.196152423 Simplifying p + -9 = 5.196152423 Reorder the terms: -9 + p = 5.196152423 Solving -9 + p = 5.196152423 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + p = 5.196152423 + 9 Combine like terms: -9 + 9 = 0 0 + p = 5.196152423 + 9 p = 5.196152423 + 9 Combine like terms: 5.196152423 + 9 = 14.196152423 p = 14.196152423 Simplifying p = 14.196152423Subproblem 2
p + -9 = -5.196152423 Simplifying p + -9 = -5.196152423 Reorder the terms: -9 + p = -5.196152423 Solving -9 + p = -5.196152423 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + p = -5.196152423 + 9 Combine like terms: -9 + 9 = 0 0 + p = -5.196152423 + 9 p = -5.196152423 + 9 Combine like terms: -5.196152423 + 9 = 3.803847577 p = 3.803847577 Simplifying p = 3.803847577Solution
The solution to the problem is based on the solutions from the subproblems. p = {14.196152423, 3.803847577}Solution
p = {14.196152423, 3.803847577}
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