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Simplifying 10p4 + -20p3 + 5p2 = 0 Reorder the terms: 5p2 + -20p3 + 10p4 = 0 Solving 5p2 + -20p3 + 10p4 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), '5p2'. 5p2(1 + -4p + 2p2) = 0 Ignore the factor 5.Subproblem 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 '(1 + -4p + 2p2)' equal to zero and attempt to solve: Simplifying 1 + -4p + 2p2 = 0 Solving 1 + -4p + 2p2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + -2p + p2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2p + -0.5 + p2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2p + p2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2p + p2 = 0 + -0.5 -2p + p2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2p + p2 = -0.5 The p term is -2p. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2p + 1 + p2 = -0.5 + 1 Reorder the terms: 1 + -2p + p2 = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 1 + -2p + p2 = 0.5 Factor a perfect square on the left side: (p + -1)(p + -1) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (p + -1) equal to 0.707106781 and -0.707106781.Subproblem 1
p + -1 = 0.707106781 Simplifying p + -1 = 0.707106781 Reorder the terms: -1 + p = 0.707106781 Solving -1 + p = 0.707106781 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = 0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + p = 0.707106781 + 1 p = 0.707106781 + 1 Combine like terms: 0.707106781 + 1 = 1.707106781 p = 1.707106781 Simplifying p = 1.707106781Subproblem 2
p + -1 = -0.707106781 Simplifying p + -1 = -0.707106781 Reorder the terms: -1 + p = -0.707106781 Solving -1 + p = -0.707106781 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = -0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + p = -0.707106781 + 1 p = -0.707106781 + 1 Combine like terms: -0.707106781 + 1 = 0.292893219 p = 0.292893219 Simplifying p = 0.292893219Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.707106781, 0.292893219}Solution
p = {0, 1.707106781, 0.292893219}
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