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Simplifying 81p4 + -126p3 + -49p2 = 0 Reorder the terms: -49p2 + -126p3 + 81p4 = 0 Solving -49p2 + -126p3 + 81p4 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), 'p2'. p2(-49 + -126p + 81p2) = 0Subproblem 1
Set the factor 'p2' equal to zero and attempt to solve: Simplifying p2 = 0 Solving p2 = 0 Move all terms containing p to the left, all other terms to the right. Simplifying p2 = 0 Take the square root of each side: p = {0}Subproblem 2
Set the factor '(-49 + -126p + 81p2)' equal to zero and attempt to solve: Simplifying -49 + -126p + 81p2 = 0 Solving -49 + -126p + 81p2 = 0 Begin completing the square. Divide all terms by 81 the coefficient of the squared term: Divide each side by '81'. -0.6049382716 + -1.555555556p + p2 = 0 Move the constant term to the right: Add '0.6049382716' to each side of the equation. -0.6049382716 + -1.555555556p + 0.6049382716 + p2 = 0 + 0.6049382716 Reorder the terms: -0.6049382716 + 0.6049382716 + -1.555555556p + p2 = 0 + 0.6049382716 Combine like terms: -0.6049382716 + 0.6049382716 = 0.0000000000 0.0000000000 + -1.555555556p + p2 = 0 + 0.6049382716 -1.555555556p + p2 = 0 + 0.6049382716 Combine like terms: 0 + 0.6049382716 = 0.6049382716 -1.555555556p + p2 = 0.6049382716 The p term is -1.555555556p. Take half its coefficient (-0.777777778). Square it (0.6049382720) and add it to both sides. Add '0.6049382720' to each side of the equation. -1.555555556p + 0.6049382720 + p2 = 0.6049382716 + 0.6049382720 Reorder the terms: 0.6049382720 + -1.555555556p + p2 = 0.6049382716 + 0.6049382720 Combine like terms: 0.6049382716 + 0.6049382720 = 1.2098765436 0.6049382720 + -1.555555556p + p2 = 1.2098765436 Factor a perfect square on the left side: (p + -0.777777778)(p + -0.777777778) = 1.2098765436 Calculate the square root of the right side: 1.099943882 Break this problem into two subproblems by setting (p + -0.777777778) equal to 1.099943882 and -1.099943882.Subproblem 1
p + -0.777777778 = 1.099943882 Simplifying p + -0.777777778 = 1.099943882 Reorder the terms: -0.777777778 + p = 1.099943882 Solving -0.777777778 + p = 1.099943882 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.777777778' to each side of the equation. -0.777777778 + 0.777777778 + p = 1.099943882 + 0.777777778 Combine like terms: -0.777777778 + 0.777777778 = 0.000000000 0.000000000 + p = 1.099943882 + 0.777777778 p = 1.099943882 + 0.777777778 Combine like terms: 1.099943882 + 0.777777778 = 1.87772166 p = 1.87772166 Simplifying p = 1.87772166Subproblem 2
p + -0.777777778 = -1.099943882 Simplifying p + -0.777777778 = -1.099943882 Reorder the terms: -0.777777778 + p = -1.099943882 Solving -0.777777778 + p = -1.099943882 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.777777778' to each side of the equation. -0.777777778 + 0.777777778 + p = -1.099943882 + 0.777777778 Combine like terms: -0.777777778 + 0.777777778 = 0.000000000 0.000000000 + p = -1.099943882 + 0.777777778 p = -1.099943882 + 0.777777778 Combine like terms: -1.099943882 + 0.777777778 = -0.322166104 p = -0.322166104 Simplifying p = -0.322166104Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.87772166, -0.322166104}Solution
p = {0, 1.87772166, -0.322166104}
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