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Simplifying 3(2p + 7) + 8p(4p + -9) = 0 Reorder the terms: 3(7 + 2p) + 8p(4p + -9) = 0 (7 * 3 + 2p * 3) + 8p(4p + -9) = 0 (21 + 6p) + 8p(4p + -9) = 0 Reorder the terms: 21 + 6p + 8p(-9 + 4p) = 0 21 + 6p + (-9 * 8p + 4p * 8p) = 0 21 + 6p + (-72p + 32p2) = 0 Combine like terms: 6p + -72p = -66p 21 + -66p + 32p2 = 0 Solving 21 + -66p + 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'. 0.65625 + -2.0625p + p2 = 0 Move the constant term to the right: Add '-0.65625' to each side of the equation. 0.65625 + -2.0625p + -0.65625 + p2 = 0 + -0.65625 Reorder the terms: 0.65625 + -0.65625 + -2.0625p + p2 = 0 + -0.65625 Combine like terms: 0.65625 + -0.65625 = 0.00000 0.00000 + -2.0625p + p2 = 0 + -0.65625 -2.0625p + p2 = 0 + -0.65625 Combine like terms: 0 + -0.65625 = -0.65625 -2.0625p + p2 = -0.65625 The p term is -2.0625p. Take half its coefficient (-1.03125). Square it (1.063476563) and add it to both sides. Add '1.063476563' to each side of the equation. -2.0625p + 1.063476563 + p2 = -0.65625 + 1.063476563 Reorder the terms: 1.063476563 + -2.0625p + p2 = -0.65625 + 1.063476563 Combine like terms: -0.65625 + 1.063476563 = 0.407226563 1.063476563 + -2.0625p + p2 = 0.407226563 Factor a perfect square on the left side: (p + -1.03125)(p + -1.03125) = 0.407226563 Calculate the square root of the right side: 0.638143058 Break this problem into two subproblems by setting (p + -1.03125) equal to 0.638143058 and -0.638143058.Subproblem 1
p + -1.03125 = 0.638143058 Simplifying p + -1.03125 = 0.638143058 Reorder the terms: -1.03125 + p = 0.638143058 Solving -1.03125 + p = 0.638143058 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + p = 0.638143058 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + p = 0.638143058 + 1.03125 p = 0.638143058 + 1.03125 Combine like terms: 0.638143058 + 1.03125 = 1.669393058 p = 1.669393058 Simplifying p = 1.669393058Subproblem 2
p + -1.03125 = -0.638143058 Simplifying p + -1.03125 = -0.638143058 Reorder the terms: -1.03125 + p = -0.638143058 Solving -1.03125 + p = -0.638143058 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + p = -0.638143058 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + p = -0.638143058 + 1.03125 p = -0.638143058 + 1.03125 Combine like terms: -0.638143058 + 1.03125 = 0.393106942 p = 0.393106942 Simplifying p = 0.393106942Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.669393058, 0.393106942}
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