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Simplifying (p + 2)(p + 2) = p + 20 Reorder the terms: (2 + p)(p + 2) = p + 20 Reorder the terms: (2 + p)(2 + p) = p + 20 Multiply (2 + p) * (2 + p) (2(2 + p) + p(2 + p)) = p + 20 ((2 * 2 + p * 2) + p(2 + p)) = p + 20 ((4 + 2p) + p(2 + p)) = p + 20 (4 + 2p + (2 * p + p * p)) = p + 20 (4 + 2p + (2p + p2)) = p + 20 Combine like terms: 2p + 2p = 4p (4 + 4p + p2) = p + 20 Reorder the terms: 4 + 4p + p2 = 20 + p Solving 4 + 4p + p2 = 20 + p Solving for variable 'p'. Reorder the terms: 4 + -20 + 4p + -1p + p2 = 20 + p + -20 + -1p Combine like terms: 4 + -20 = -16 -16 + 4p + -1p + p2 = 20 + p + -20 + -1p Combine like terms: 4p + -1p = 3p -16 + 3p + p2 = 20 + p + -20 + -1p Reorder the terms: -16 + 3p + p2 = 20 + -20 + p + -1p Combine like terms: 20 + -20 = 0 -16 + 3p + p2 = 0 + p + -1p -16 + 3p + p2 = p + -1p Combine like terms: p + -1p = 0 -16 + 3p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '16' to each side of the equation. -16 + 3p + 16 + p2 = 0 + 16 Reorder the terms: -16 + 16 + 3p + p2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 3p + p2 = 0 + 16 3p + p2 = 0 + 16 Combine like terms: 0 + 16 = 16 3p + p2 = 16 The p term is 3p. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3p + 2.25 + p2 = 16 + 2.25 Reorder the terms: 2.25 + 3p + p2 = 16 + 2.25 Combine like terms: 16 + 2.25 = 18.25 2.25 + 3p + p2 = 18.25 Factor a perfect square on the left side: (p + 1.5)(p + 1.5) = 18.25 Calculate the square root of the right side: 4.272001873 Break this problem into two subproblems by setting (p + 1.5) equal to 4.272001873 and -4.272001873.Subproblem 1
p + 1.5 = 4.272001873 Simplifying p + 1.5 = 4.272001873 Reorder the terms: 1.5 + p = 4.272001873 Solving 1.5 + p = 4.272001873 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = 4.272001873 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = 4.272001873 + -1.5 p = 4.272001873 + -1.5 Combine like terms: 4.272001873 + -1.5 = 2.772001873 p = 2.772001873 Simplifying p = 2.772001873Subproblem 2
p + 1.5 = -4.272001873 Simplifying p + 1.5 = -4.272001873 Reorder the terms: 1.5 + p = -4.272001873 Solving 1.5 + p = -4.272001873 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = -4.272001873 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = -4.272001873 + -1.5 p = -4.272001873 + -1.5 Combine like terms: -4.272001873 + -1.5 = -5.772001873 p = -5.772001873 Simplifying p = -5.772001873Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.772001873, -5.772001873}
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