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Simplifying 32p2 + -31p + -40 = 0 Reorder the terms: -40 + -31p + 32p2 = 0 Solving -40 + -31p + 32p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -1.25 + -0.96875p + p2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -0.96875p + 1.25 + p2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -0.96875p + p2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -0.96875p + p2 = 0 + 1.25 -0.96875p + p2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -0.96875p + p2 = 1.25 The p term is -0.96875p. Take half its coefficient (-0.484375). Square it (0.2346191406) and add it to both sides. Add '0.2346191406' to each side of the equation. -0.96875p + 0.2346191406 + p2 = 1.25 + 0.2346191406 Reorder the terms: 0.2346191406 + -0.96875p + p2 = 1.25 + 0.2346191406 Combine like terms: 1.25 + 0.2346191406 = 1.4846191406 0.2346191406 + -0.96875p + p2 = 1.4846191406 Factor a perfect square on the left side: (p + -0.484375)(p + -0.484375) = 1.4846191406 Calculate the square root of the right side: 1.218449482 Break this problem into two subproblems by setting (p + -0.484375) equal to 1.218449482 and -1.218449482.Subproblem 1
p + -0.484375 = 1.218449482 Simplifying p + -0.484375 = 1.218449482 Reorder the terms: -0.484375 + p = 1.218449482 Solving -0.484375 + p = 1.218449482 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.484375' to each side of the equation. -0.484375 + 0.484375 + p = 1.218449482 + 0.484375 Combine like terms: -0.484375 + 0.484375 = 0.000000 0.000000 + p = 1.218449482 + 0.484375 p = 1.218449482 + 0.484375 Combine like terms: 1.218449482 + 0.484375 = 1.702824482 p = 1.702824482 Simplifying p = 1.702824482Subproblem 2
p + -0.484375 = -1.218449482 Simplifying p + -0.484375 = -1.218449482 Reorder the terms: -0.484375 + p = -1.218449482 Solving -0.484375 + p = -1.218449482 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.484375' to each side of the equation. -0.484375 + 0.484375 + p = -1.218449482 + 0.484375 Combine like terms: -0.484375 + 0.484375 = 0.000000 0.000000 + p = -1.218449482 + 0.484375 p = -1.218449482 + 0.484375 Combine like terms: -1.218449482 + 0.484375 = -0.734074482 p = -0.734074482 Simplifying p = -0.734074482Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.702824482, -0.734074482}
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