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Simplifying 2p(p2 + -6p + 1) = 0 Reorder the terms: 2p(1 + -6p + p2) = 0 (1 * 2p + -6p * 2p + p2 * 2p) = 0 (2p + -12p2 + 2p3) = 0 Solving 2p + -12p2 + 2p3 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), '2p'. 2p(1 + -6p + p2) = 0 Ignore the factor 2.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 '(1 + -6p + p2)' equal to zero and attempt to solve: Simplifying 1 + -6p + p2 = 0 Solving 1 + -6p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -6p + -1 + p2 = 0 + -1 Reorder the terms: 1 + -1 + -6p + p2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -6p + p2 = 0 + -1 -6p + p2 = 0 + -1 Combine like terms: 0 + -1 = -1 -6p + p2 = -1 The p term is -6p. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6p + 9 + p2 = -1 + 9 Reorder the terms: 9 + -6p + p2 = -1 + 9 Combine like terms: -1 + 9 = 8 9 + -6p + p2 = 8 Factor a perfect square on the left side: (p + -3)(p + -3) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (p + -3) equal to 2.828427125 and -2.828427125.Subproblem 1
p + -3 = 2.828427125 Simplifying p + -3 = 2.828427125 Reorder the terms: -3 + p = 2.828427125 Solving -3 + p = 2.828427125 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + p = 2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + p = 2.828427125 + 3 p = 2.828427125 + 3 Combine like terms: 2.828427125 + 3 = 5.828427125 p = 5.828427125 Simplifying p = 5.828427125Subproblem 2
p + -3 = -2.828427125 Simplifying p + -3 = -2.828427125 Reorder the terms: -3 + p = -2.828427125 Solving -3 + p = -2.828427125 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + p = -2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + p = -2.828427125 + 3 p = -2.828427125 + 3 Combine like terms: -2.828427125 + 3 = 0.171572875 p = 0.171572875 Simplifying p = 0.171572875Solution
The solution to the problem is based on the solutions from the subproblems. p = {5.828427125, 0.171572875}Solution
p = {0, 5.828427125, 0.171572875}
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