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Simplifying -4p2 + 12p + 32 = -5p Reorder the terms: 32 + 12p + -4p2 = -5p Solving 32 + 12p + -4p2 = -5p Solving for variable 'p'. Reorder the terms: 32 + 12p + 5p + -4p2 = -5p + 5p Combine like terms: 12p + 5p = 17p 32 + 17p + -4p2 = -5p + 5p Combine like terms: -5p + 5p = 0 32 + 17p + -4p2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -8 + -4.25p + p2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -4.25p + 8 + p2 = 0 + 8 Reorder the terms: -8 + 8 + -4.25p + p2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -4.25p + p2 = 0 + 8 -4.25p + p2 = 0 + 8 Combine like terms: 0 + 8 = 8 -4.25p + p2 = 8 The p term is -4.25p. Take half its coefficient (-2.125). Square it (4.515625) and add it to both sides. Add '4.515625' to each side of the equation. -4.25p + 4.515625 + p2 = 8 + 4.515625 Reorder the terms: 4.515625 + -4.25p + p2 = 8 + 4.515625 Combine like terms: 8 + 4.515625 = 12.515625 4.515625 + -4.25p + p2 = 12.515625 Factor a perfect square on the left side: (p + -2.125)(p + -2.125) = 12.515625 Calculate the square root of the right side: 3.537742925 Break this problem into two subproblems by setting (p + -2.125) equal to 3.537742925 and -3.537742925.Subproblem 1
p + -2.125 = 3.537742925 Simplifying p + -2.125 = 3.537742925 Reorder the terms: -2.125 + p = 3.537742925 Solving -2.125 + p = 3.537742925 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + p = 3.537742925 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + p = 3.537742925 + 2.125 p = 3.537742925 + 2.125 Combine like terms: 3.537742925 + 2.125 = 5.662742925 p = 5.662742925 Simplifying p = 5.662742925Subproblem 2
p + -2.125 = -3.537742925 Simplifying p + -2.125 = -3.537742925 Reorder the terms: -2.125 + p = -3.537742925 Solving -2.125 + p = -3.537742925 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + p = -3.537742925 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + p = -3.537742925 + 2.125 p = -3.537742925 + 2.125 Combine like terms: -3.537742925 + 2.125 = -1.412742925 p = -1.412742925 Simplifying p = -1.412742925Solution
The solution to the problem is based on the solutions from the subproblems. p = {5.662742925, -1.412742925}
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