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Simplifying 90p3 + -80p2 + -40p = 0 Reorder the terms: -40p + -80p2 + 90p3 = 0 Solving -40p + -80p2 + 90p3 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), '10p'. 10p(-4 + -8p + 9p2) = 0 Ignore the factor 10.Subproblem 1
Set the factor 'p' equal to zero and attempt to solve: Simplifying p = 0 Solving p = 0 Move all terms containing p to the left, all other terms to the right. Simplifying p = 0Subproblem 2
Set the factor '(-4 + -8p + 9p2)' equal to zero and attempt to solve: Simplifying -4 + -8p + 9p2 = 0 Solving -4 + -8p + 9p2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.4444444444 + -0.8888888889p + p2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + -0.8888888889p + 0.4444444444 + p2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + -0.8888888889p + p2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + -0.8888888889p + p2 = 0 + 0.4444444444 -0.8888888889p + p2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 -0.8888888889p + p2 = 0.4444444444 The p term is -0.8888888889p. Take half its coefficient (-0.4444444445). Square it (0.1975308642) and add it to both sides. Add '0.1975308642' to each side of the equation. -0.8888888889p + 0.1975308642 + p2 = 0.4444444444 + 0.1975308642 Reorder the terms: 0.1975308642 + -0.8888888889p + p2 = 0.4444444444 + 0.1975308642 Combine like terms: 0.4444444444 + 0.1975308642 = 0.6419753086 0.1975308642 + -0.8888888889p + p2 = 0.6419753086 Factor a perfect square on the left side: (p + -0.4444444445)(p + -0.4444444445) = 0.6419753086 Calculate the square root of the right side: 0.801233617 Break this problem into two subproblems by setting (p + -0.4444444445) equal to 0.801233617 and -0.801233617.Subproblem 1
p + -0.4444444445 = 0.801233617 Simplifying p + -0.4444444445 = 0.801233617 Reorder the terms: -0.4444444445 + p = 0.801233617 Solving -0.4444444445 + p = 0.801233617 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.4444444445' to each side of the equation. -0.4444444445 + 0.4444444445 + p = 0.801233617 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + p = 0.801233617 + 0.4444444445 p = 0.801233617 + 0.4444444445 Combine like terms: 0.801233617 + 0.4444444445 = 1.2456780615 p = 1.2456780615 Simplifying p = 1.2456780615Subproblem 2
p + -0.4444444445 = -0.801233617 Simplifying p + -0.4444444445 = -0.801233617 Reorder the terms: -0.4444444445 + p = -0.801233617 Solving -0.4444444445 + p = -0.801233617 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.4444444445' to each side of the equation. -0.4444444445 + 0.4444444445 + p = -0.801233617 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + p = -0.801233617 + 0.4444444445 p = -0.801233617 + 0.4444444445 Combine like terms: -0.801233617 + 0.4444444445 = -0.3567891725 p = -0.3567891725 Simplifying p = -0.3567891725Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.2456780615, -0.3567891725}Solution
p = {0, 1.2456780615, -0.3567891725}
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