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Simplifying (p + 3)(p + -5) = 2 Reorder the terms: (3 + p)(p + -5) = 2 Reorder the terms: (3 + p)(-5 + p) = 2 Multiply (3 + p) * (-5 + p) (3(-5 + p) + p(-5 + p)) = 2 ((-5 * 3 + p * 3) + p(-5 + p)) = 2 ((-15 + 3p) + p(-5 + p)) = 2 (-15 + 3p + (-5 * p + p * p)) = 2 (-15 + 3p + (-5p + p2)) = 2 Combine like terms: 3p + -5p = -2p (-15 + -2p + p2) = 2 Solving -15 + -2p + p2 = 2 Solving for variable 'p'. Reorder the terms: -15 + -2 + -2p + p2 = 2 + -2 Combine like terms: -15 + -2 = -17 -17 + -2p + p2 = 2 + -2 Combine like terms: 2 + -2 = 0 -17 + -2p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '17' to each side of the equation. -17 + -2p + 17 + p2 = 0 + 17 Reorder the terms: -17 + 17 + -2p + p2 = 0 + 17 Combine like terms: -17 + 17 = 0 0 + -2p + p2 = 0 + 17 -2p + p2 = 0 + 17 Combine like terms: 0 + 17 = 17 -2p + p2 = 17 The p term is -2p. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2p + 1 + p2 = 17 + 1 Reorder the terms: 1 + -2p + p2 = 17 + 1 Combine like terms: 17 + 1 = 18 1 + -2p + p2 = 18 Factor a perfect square on the left side: (p + -1)(p + -1) = 18 Calculate the square root of the right side: 4.242640687 Break this problem into two subproblems by setting (p + -1) equal to 4.242640687 and -4.242640687.Subproblem 1
p + -1 = 4.242640687 Simplifying p + -1 = 4.242640687 Reorder the terms: -1 + p = 4.242640687 Solving -1 + p = 4.242640687 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = 4.242640687 + 1 Combine like terms: -1 + 1 = 0 0 + p = 4.242640687 + 1 p = 4.242640687 + 1 Combine like terms: 4.242640687 + 1 = 5.242640687 p = 5.242640687 Simplifying p = 5.242640687Subproblem 2
p + -1 = -4.242640687 Simplifying p + -1 = -4.242640687 Reorder the terms: -1 + p = -4.242640687 Solving -1 + p = -4.242640687 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = -4.242640687 + 1 Combine like terms: -1 + 1 = 0 0 + p = -4.242640687 + 1 p = -4.242640687 + 1 Combine like terms: -4.242640687 + 1 = -3.242640687 p = -3.242640687 Simplifying p = -3.242640687Solution
The solution to the problem is based on the solutions from the subproblems. p = {5.242640687, -3.242640687}
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