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Simplifying (5p + 2)(p + 4) = 1 Reorder the terms: (2 + 5p)(p + 4) = 1 Reorder the terms: (2 + 5p)(4 + p) = 1 Multiply (2 + 5p) * (4 + p) (2(4 + p) + 5p * (4 + p)) = 1 ((4 * 2 + p * 2) + 5p * (4 + p)) = 1 ((8 + 2p) + 5p * (4 + p)) = 1 (8 + 2p + (4 * 5p + p * 5p)) = 1 (8 + 2p + (20p + 5p2)) = 1 Combine like terms: 2p + 20p = 22p (8 + 22p + 5p2) = 1 Solving 8 + 22p + 5p2 = 1 Solving for variable 'p'. Reorder the terms: 8 + -1 + 22p + 5p2 = 1 + -1 Combine like terms: 8 + -1 = 7 7 + 22p + 5p2 = 1 + -1 Combine like terms: 1 + -1 = 0 7 + 22p + 5p2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 1.4 + 4.4p + p2 = 0 Move the constant term to the right: Add '-1.4' to each side of the equation. 1.4 + 4.4p + -1.4 + p2 = 0 + -1.4 Reorder the terms: 1.4 + -1.4 + 4.4p + p2 = 0 + -1.4 Combine like terms: 1.4 + -1.4 = 0.0 0.0 + 4.4p + p2 = 0 + -1.4 4.4p + p2 = 0 + -1.4 Combine like terms: 0 + -1.4 = -1.4 4.4p + p2 = -1.4 The p term is 4.4p. Take half its coefficient (2.2). Square it (4.84) and add it to both sides. Add '4.84' to each side of the equation. 4.4p + 4.84 + p2 = -1.4 + 4.84 Reorder the terms: 4.84 + 4.4p + p2 = -1.4 + 4.84 Combine like terms: -1.4 + 4.84 = 3.44 4.84 + 4.4p + p2 = 3.44 Factor a perfect square on the left side: (p + 2.2)(p + 2.2) = 3.44 Calculate the square root of the right side: 1.854723699 Break this problem into two subproblems by setting (p + 2.2) equal to 1.854723699 and -1.854723699.Subproblem 1
p + 2.2 = 1.854723699 Simplifying p + 2.2 = 1.854723699 Reorder the terms: 2.2 + p = 1.854723699 Solving 2.2 + p = 1.854723699 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.2' to each side of the equation. 2.2 + -2.2 + p = 1.854723699 + -2.2 Combine like terms: 2.2 + -2.2 = 0.0 0.0 + p = 1.854723699 + -2.2 p = 1.854723699 + -2.2 Combine like terms: 1.854723699 + -2.2 = -0.345276301 p = -0.345276301 Simplifying p = -0.345276301Subproblem 2
p + 2.2 = -1.854723699 Simplifying p + 2.2 = -1.854723699 Reorder the terms: 2.2 + p = -1.854723699 Solving 2.2 + p = -1.854723699 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.2' to each side of the equation. 2.2 + -2.2 + p = -1.854723699 + -2.2 Combine like terms: 2.2 + -2.2 = 0.0 0.0 + p = -1.854723699 + -2.2 p = -1.854723699 + -2.2 Combine like terms: -1.854723699 + -2.2 = -4.054723699 p = -4.054723699 Simplifying p = -4.054723699Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.345276301, -4.054723699}
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