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Simplifying 3p(3p + -8) + -8 = 9p(p + -3) * 4 Reorder the terms: 3p(-8 + 3p) + -8 = 9p(p + -3) * 4 (-8 * 3p + 3p * 3p) + -8 = 9p(p + -3) * 4 (-24p + 9p2) + -8 = 9p(p + -3) * 4 Reorder the terms: -8 + -24p + 9p2 = 9p(p + -3) * 4 Reorder the terms: -8 + -24p + 9p2 = 9p(-3 + p) * 4 Reorder the terms for easier multiplication: -8 + -24p + 9p2 = 9 * 4p(-3 + p) Multiply 9 * 4 -8 + -24p + 9p2 = 36p(-3 + p) -8 + -24p + 9p2 = (-3 * 36p + p * 36p) -8 + -24p + 9p2 = (-108p + 36p2) Solving -8 + -24p + 9p2 = -108p + 36p2 Solving for variable 'p'. Reorder the terms: -8 + -24p + 108p + 9p2 + -36p2 = -108p + 36p2 + 108p + -36p2 Combine like terms: -24p + 108p = 84p -8 + 84p + 9p2 + -36p2 = -108p + 36p2 + 108p + -36p2 Combine like terms: 9p2 + -36p2 = -27p2 -8 + 84p + -27p2 = -108p + 36p2 + 108p + -36p2 Reorder the terms: -8 + 84p + -27p2 = -108p + 108p + 36p2 + -36p2 Combine like terms: -108p + 108p = 0 -8 + 84p + -27p2 = 0 + 36p2 + -36p2 -8 + 84p + -27p2 = 36p2 + -36p2 Combine like terms: 36p2 + -36p2 = 0 -8 + 84p + -27p2 = 0 Begin completing the square. Divide all terms by -27 the coefficient of the squared term: Divide each side by '-27'. 0.2962962963 + -3.111111111p + p2 = 0 Move the constant term to the right: Add '-0.2962962963' to each side of the equation. 0.2962962963 + -3.111111111p + -0.2962962963 + p2 = 0 + -0.2962962963 Reorder the terms: 0.2962962963 + -0.2962962963 + -3.111111111p + p2 = 0 + -0.2962962963 Combine like terms: 0.2962962963 + -0.2962962963 = 0.0000000000 0.0000000000 + -3.111111111p + p2 = 0 + -0.2962962963 -3.111111111p + p2 = 0 + -0.2962962963 Combine like terms: 0 + -0.2962962963 = -0.2962962963 -3.111111111p + p2 = -0.2962962963 The p term is -3.111111111p. Take half its coefficient (-1.555555556). Square it (2.419753088) and add it to both sides. Add '2.419753088' to each side of the equation. -3.111111111p + 2.419753088 + p2 = -0.2962962963 + 2.419753088 Reorder the terms: 2.419753088 + -3.111111111p + p2 = -0.2962962963 + 2.419753088 Combine like terms: -0.2962962963 + 2.419753088 = 2.1234567917 2.419753088 + -3.111111111p + p2 = 2.1234567917 Factor a perfect square on the left side: (p + -1.555555556)(p + -1.555555556) = 2.1234567917 Calculate the square root of the right side: 1.457208561 Break this problem into two subproblems by setting (p + -1.555555556) equal to 1.457208561 and -1.457208561.Subproblem 1
p + -1.555555556 = 1.457208561 Simplifying p + -1.555555556 = 1.457208561 Reorder the terms: -1.555555556 + p = 1.457208561 Solving -1.555555556 + p = 1.457208561 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.555555556' to each side of the equation. -1.555555556 + 1.555555556 + p = 1.457208561 + 1.555555556 Combine like terms: -1.555555556 + 1.555555556 = 0.000000000 0.000000000 + p = 1.457208561 + 1.555555556 p = 1.457208561 + 1.555555556 Combine like terms: 1.457208561 + 1.555555556 = 3.012764117 p = 3.012764117 Simplifying p = 3.012764117Subproblem 2
p + -1.555555556 = -1.457208561 Simplifying p + -1.555555556 = -1.457208561 Reorder the terms: -1.555555556 + p = -1.457208561 Solving -1.555555556 + p = -1.457208561 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.555555556' to each side of the equation. -1.555555556 + 1.555555556 + p = -1.457208561 + 1.555555556 Combine like terms: -1.555555556 + 1.555555556 = 0.000000000 0.000000000 + p = -1.457208561 + 1.555555556 p = -1.457208561 + 1.555555556 Combine like terms: -1.457208561 + 1.555555556 = 0.098346995 p = 0.098346995 Simplifying p = 0.098346995Solution
The solution to the problem is based on the solutions from the subproblems. p = {3.012764117, 0.098346995}
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