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Simplifying 6(p + 2) = 5(p + -3) * p + 27 Reorder the terms: 6(2 + p) = 5(p + -3) * p + 27 (2 * 6 + p * 6) = 5(p + -3) * p + 27 (12 + 6p) = 5(p + -3) * p + 27 Reorder the terms: 12 + 6p = 5(-3 + p) * p + 27 Reorder the terms for easier multiplication: 12 + 6p = 5p(-3 + p) + 27 12 + 6p = (-3 * 5p + p * 5p) + 27 12 + 6p = (-15p + 5p2) + 27 Reorder the terms: 12 + 6p = 27 + -15p + 5p2 Solving 12 + 6p = 27 + -15p + 5p2 Solving for variable 'p'. Reorder the terms: 12 + -27 + 6p + 15p + -5p2 = 27 + -15p + 5p2 + -27 + 15p + -5p2 Combine like terms: 12 + -27 = -15 -15 + 6p + 15p + -5p2 = 27 + -15p + 5p2 + -27 + 15p + -5p2 Combine like terms: 6p + 15p = 21p -15 + 21p + -5p2 = 27 + -15p + 5p2 + -27 + 15p + -5p2 Reorder the terms: -15 + 21p + -5p2 = 27 + -27 + -15p + 15p + 5p2 + -5p2 Combine like terms: 27 + -27 = 0 -15 + 21p + -5p2 = 0 + -15p + 15p + 5p2 + -5p2 -15 + 21p + -5p2 = -15p + 15p + 5p2 + -5p2 Combine like terms: -15p + 15p = 0 -15 + 21p + -5p2 = 0 + 5p2 + -5p2 -15 + 21p + -5p2 = 5p2 + -5p2 Combine like terms: 5p2 + -5p2 = 0 -15 + 21p + -5p2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 3 + -4.2p + p2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -4.2p + -3 + p2 = 0 + -3 Reorder the terms: 3 + -3 + -4.2p + p2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -4.2p + p2 = 0 + -3 -4.2p + p2 = 0 + -3 Combine like terms: 0 + -3 = -3 -4.2p + p2 = -3 The p term is -4.2p. Take half its coefficient (-2.1). Square it (4.41) and add it to both sides. Add '4.41' to each side of the equation. -4.2p + 4.41 + p2 = -3 + 4.41 Reorder the terms: 4.41 + -4.2p + p2 = -3 + 4.41 Combine like terms: -3 + 4.41 = 1.41 4.41 + -4.2p + p2 = 1.41 Factor a perfect square on the left side: (p + -2.1)(p + -2.1) = 1.41 Calculate the square root of the right side: 1.187434209 Break this problem into two subproblems by setting (p + -2.1) equal to 1.187434209 and -1.187434209.Subproblem 1
p + -2.1 = 1.187434209 Simplifying p + -2.1 = 1.187434209 Reorder the terms: -2.1 + p = 1.187434209 Solving -2.1 + p = 1.187434209 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.1' to each side of the equation. -2.1 + 2.1 + p = 1.187434209 + 2.1 Combine like terms: -2.1 + 2.1 = 0.0 0.0 + p = 1.187434209 + 2.1 p = 1.187434209 + 2.1 Combine like terms: 1.187434209 + 2.1 = 3.287434209 p = 3.287434209 Simplifying p = 3.287434209Subproblem 2
p + -2.1 = -1.187434209 Simplifying p + -2.1 = -1.187434209 Reorder the terms: -2.1 + p = -1.187434209 Solving -2.1 + p = -1.187434209 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.1' to each side of the equation. -2.1 + 2.1 + p = -1.187434209 + 2.1 Combine like terms: -2.1 + 2.1 = 0.0 0.0 + p = -1.187434209 + 2.1 p = -1.187434209 + 2.1 Combine like terms: -1.187434209 + 2.1 = 0.912565791 p = 0.912565791 Simplifying p = 0.912565791Solution
The solution to the problem is based on the solutions from the subproblems. p = {3.287434209, 0.912565791}
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