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Simplifying -35 + -1 = -7p(8p + 5) Combine like terms: -35 + -1 = -36 -36 = -7p(8p + 5) Reorder the terms: -36 = -7p(5 + 8p) -36 = (5 * -7p + 8p * -7p) -36 = (-35p + -56p2) Solving -36 = -35p + -56p2 Solving for variable 'p'. Reorder the terms: -36 + 35p + 56p2 = -35p + 35p + -56p2 + 56p2 Combine like terms: -35p + 35p = 0 -36 + 35p + 56p2 = 0 + -56p2 + 56p2 -36 + 35p + 56p2 = -56p2 + 56p2 Combine like terms: -56p2 + 56p2 = 0 -36 + 35p + 56p2 = 0 Begin completing the square. Divide all terms by 56 the coefficient of the squared term: Divide each side by '56'. -0.6428571429 + 0.625p + p2 = 0 Move the constant term to the right: Add '0.6428571429' to each side of the equation. -0.6428571429 + 0.625p + 0.6428571429 + p2 = 0 + 0.6428571429 Reorder the terms: -0.6428571429 + 0.6428571429 + 0.625p + p2 = 0 + 0.6428571429 Combine like terms: -0.6428571429 + 0.6428571429 = 0.0000000000 0.0000000000 + 0.625p + p2 = 0 + 0.6428571429 0.625p + p2 = 0 + 0.6428571429 Combine like terms: 0 + 0.6428571429 = 0.6428571429 0.625p + p2 = 0.6428571429 The p term is 0.625p. Take half its coefficient (0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. 0.625p + 0.09765625 + p2 = 0.6428571429 + 0.09765625 Reorder the terms: 0.09765625 + 0.625p + p2 = 0.6428571429 + 0.09765625 Combine like terms: 0.6428571429 + 0.09765625 = 0.7405133929 0.09765625 + 0.625p + p2 = 0.7405133929 Factor a perfect square on the left side: (p + 0.3125)(p + 0.3125) = 0.7405133929 Calculate the square root of the right side: 0.860530879 Break this problem into two subproblems by setting (p + 0.3125) equal to 0.860530879 and -0.860530879.Subproblem 1
p + 0.3125 = 0.860530879 Simplifying p + 0.3125 = 0.860530879 Reorder the terms: 0.3125 + p = 0.860530879 Solving 0.3125 + p = 0.860530879 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + p = 0.860530879 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + p = 0.860530879 + -0.3125 p = 0.860530879 + -0.3125 Combine like terms: 0.860530879 + -0.3125 = 0.548030879 p = 0.548030879 Simplifying p = 0.548030879Subproblem 2
p + 0.3125 = -0.860530879 Simplifying p + 0.3125 = -0.860530879 Reorder the terms: 0.3125 + p = -0.860530879 Solving 0.3125 + p = -0.860530879 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + p = -0.860530879 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + p = -0.860530879 + -0.3125 p = -0.860530879 + -0.3125 Combine like terms: -0.860530879 + -0.3125 = -1.173030879 p = -1.173030879 Simplifying p = -1.173030879Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.548030879, -1.173030879}
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