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Simplifying 500 = 8p2 + 128p + -640 Reorder the terms: 500 = -640 + 128p + 8p2 Solving 500 = -640 + 128p + 8p2 Solving for variable 'p'. Combine like terms: 500 + 640 = 1140 1140 + -128p + -8p2 = -640 + 128p + 8p2 + 640 + -128p + -8p2 Reorder the terms: 1140 + -128p + -8p2 = -640 + 640 + 128p + -128p + 8p2 + -8p2 Combine like terms: -640 + 640 = 0 1140 + -128p + -8p2 = 0 + 128p + -128p + 8p2 + -8p2 1140 + -128p + -8p2 = 128p + -128p + 8p2 + -8p2 Combine like terms: 128p + -128p = 0 1140 + -128p + -8p2 = 0 + 8p2 + -8p2 1140 + -128p + -8p2 = 8p2 + -8p2 Combine like terms: 8p2 + -8p2 = 0 1140 + -128p + -8p2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(285 + -32p + -2p2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(285 + -32p + -2p2)' equal to zero and attempt to solve: Simplifying 285 + -32p + -2p2 = 0 Solving 285 + -32p + -2p2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -142.5 + 16p + p2 = 0 Move the constant term to the right: Add '142.5' to each side of the equation. -142.5 + 16p + 142.5 + p2 = 0 + 142.5 Reorder the terms: -142.5 + 142.5 + 16p + p2 = 0 + 142.5 Combine like terms: -142.5 + 142.5 = 0.0 0.0 + 16p + p2 = 0 + 142.5 16p + p2 = 0 + 142.5 Combine like terms: 0 + 142.5 = 142.5 16p + p2 = 142.5 The p term is 16p. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16p + 64 + p2 = 142.5 + 64 Reorder the terms: 64 + 16p + p2 = 142.5 + 64 Combine like terms: 142.5 + 64 = 206.5 64 + 16p + p2 = 206.5 Factor a perfect square on the left side: (p + 8)(p + 8) = 206.5 Calculate the square root of the right side: 14.370107863 Break this problem into two subproblems by setting (p + 8) equal to 14.370107863 and -14.370107863.Subproblem 1
p + 8 = 14.370107863 Simplifying p + 8 = 14.370107863 Reorder the terms: 8 + p = 14.370107863 Solving 8 + p = 14.370107863 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + p = 14.370107863 + -8 Combine like terms: 8 + -8 = 0 0 + p = 14.370107863 + -8 p = 14.370107863 + -8 Combine like terms: 14.370107863 + -8 = 6.370107863 p = 6.370107863 Simplifying p = 6.370107863Subproblem 2
p + 8 = -14.370107863 Simplifying p + 8 = -14.370107863 Reorder the terms: 8 + p = -14.370107863 Solving 8 + p = -14.370107863 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + p = -14.370107863 + -8 Combine like terms: 8 + -8 = 0 0 + p = -14.370107863 + -8 p = -14.370107863 + -8 Combine like terms: -14.370107863 + -8 = -22.370107863 p = -22.370107863 Simplifying p = -22.370107863Solution
The solution to the problem is based on the solutions from the subproblems. p = {6.370107863, -22.370107863}Solution
p = {6.370107863, -22.370107863}
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