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Simplifying 3(8p2 + -5p) + -5(3p2 + -2p + 4) = 0 Reorder the terms: 3(-5p + 8p2) + -5(3p2 + -2p + 4) = 0 (-5p * 3 + 8p2 * 3) + -5(3p2 + -2p + 4) = 0 (-15p + 24p2) + -5(3p2 + -2p + 4) = 0 Reorder the terms: -15p + 24p2 + -5(4 + -2p + 3p2) = 0 -15p + 24p2 + (4 * -5 + -2p * -5 + 3p2 * -5) = 0 -15p + 24p2 + (-20 + 10p + -15p2) = 0 Reorder the terms: -20 + -15p + 10p + 24p2 + -15p2 = 0 Combine like terms: -15p + 10p = -5p -20 + -5p + 24p2 + -15p2 = 0 Combine like terms: 24p2 + -15p2 = 9p2 -20 + -5p + 9p2 = 0 Solving -20 + -5p + 9p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.222222222 + -0.5555555556p + p2 = 0 Move the constant term to the right: Add '2.222222222' to each side of the equation. -2.222222222 + -0.5555555556p + 2.222222222 + p2 = 0 + 2.222222222 Reorder the terms: -2.222222222 + 2.222222222 + -0.5555555556p + p2 = 0 + 2.222222222 Combine like terms: -2.222222222 + 2.222222222 = 0.000000000 0.000000000 + -0.5555555556p + p2 = 0 + 2.222222222 -0.5555555556p + p2 = 0 + 2.222222222 Combine like terms: 0 + 2.222222222 = 2.222222222 -0.5555555556p + p2 = 2.222222222 The p term is -0.5555555556p. Take half its coefficient (-0.2777777778). Square it (0.07716049384) and add it to both sides. Add '0.07716049384' to each side of the equation. -0.5555555556p + 0.07716049384 + p2 = 2.222222222 + 0.07716049384 Reorder the terms: 0.07716049384 + -0.5555555556p + p2 = 2.222222222 + 0.07716049384 Combine like terms: 2.222222222 + 0.07716049384 = 2.29938271584 0.07716049384 + -0.5555555556p + p2 = 2.29938271584 Factor a perfect square on the left side: (p + -0.2777777778)(p + -0.2777777778) = 2.29938271584 Calculate the square root of the right side: 1.516371563 Break this problem into two subproblems by setting (p + -0.2777777778) equal to 1.516371563 and -1.516371563.Subproblem 1
p + -0.2777777778 = 1.516371563 Simplifying p + -0.2777777778 = 1.516371563 Reorder the terms: -0.2777777778 + p = 1.516371563 Solving -0.2777777778 + p = 1.516371563 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + p = 1.516371563 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + p = 1.516371563 + 0.2777777778 p = 1.516371563 + 0.2777777778 Combine like terms: 1.516371563 + 0.2777777778 = 1.7941493408 p = 1.7941493408 Simplifying p = 1.7941493408Subproblem 2
p + -0.2777777778 = -1.516371563 Simplifying p + -0.2777777778 = -1.516371563 Reorder the terms: -0.2777777778 + p = -1.516371563 Solving -0.2777777778 + p = -1.516371563 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + p = -1.516371563 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + p = -1.516371563 + 0.2777777778 p = -1.516371563 + 0.2777777778 Combine like terms: -1.516371563 + 0.2777777778 = -1.2385937852 p = -1.2385937852 Simplifying p = -1.2385937852Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.7941493408, -1.2385937852}
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