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Simplifying (p + 5)(p + 3) = 52 Reorder the terms: (5 + p)(p + 3) = 52 Reorder the terms: (5 + p)(3 + p) = 52 Multiply (5 + p) * (3 + p) (5(3 + p) + p(3 + p)) = 52 ((3 * 5 + p * 5) + p(3 + p)) = 52 ((15 + 5p) + p(3 + p)) = 52 (15 + 5p + (3 * p + p * p)) = 52 (15 + 5p + (3p + p2)) = 52 Combine like terms: 5p + 3p = 8p (15 + 8p + p2) = 52 Solving 15 + 8p + p2 = 52 Solving for variable 'p'. Reorder the terms: 15 + -52 + 8p + p2 = 52 + -52 Combine like terms: 15 + -52 = -37 -37 + 8p + p2 = 52 + -52 Combine like terms: 52 + -52 = 0 -37 + 8p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '37' to each side of the equation. -37 + 8p + 37 + p2 = 0 + 37 Reorder the terms: -37 + 37 + 8p + p2 = 0 + 37 Combine like terms: -37 + 37 = 0 0 + 8p + p2 = 0 + 37 8p + p2 = 0 + 37 Combine like terms: 0 + 37 = 37 8p + p2 = 37 The p term is 8p. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8p + 16 + p2 = 37 + 16 Reorder the terms: 16 + 8p + p2 = 37 + 16 Combine like terms: 37 + 16 = 53 16 + 8p + p2 = 53 Factor a perfect square on the left side: (p + 4)(p + 4) = 53 Calculate the square root of the right side: 7.280109889 Break this problem into two subproblems by setting (p + 4) equal to 7.280109889 and -7.280109889.Subproblem 1
p + 4 = 7.280109889 Simplifying p + 4 = 7.280109889 Reorder the terms: 4 + p = 7.280109889 Solving 4 + p = 7.280109889 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + p = 7.280109889 + -4 Combine like terms: 4 + -4 = 0 0 + p = 7.280109889 + -4 p = 7.280109889 + -4 Combine like terms: 7.280109889 + -4 = 3.280109889 p = 3.280109889 Simplifying p = 3.280109889Subproblem 2
p + 4 = -7.280109889 Simplifying p + 4 = -7.280109889 Reorder the terms: 4 + p = -7.280109889 Solving 4 + p = -7.280109889 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + p = -7.280109889 + -4 Combine like terms: 4 + -4 = 0 0 + p = -7.280109889 + -4 p = -7.280109889 + -4 Combine like terms: -7.280109889 + -4 = -11.280109889 p = -11.280109889 Simplifying p = -11.280109889Solution
The solution to the problem is based on the solutions from the subproblems. p = {3.280109889, -11.280109889}
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