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Simplifying (6p + -2p2) + -2(4p3 + -1p) = 0 Remove parenthesis around (6p + -2p2) 6p + -2p2 + -2(4p3 + -1p) = 0 Reorder the terms: 6p + -2p2 + -2(-1p + 4p3) = 0 6p + -2p2 + (-1p * -2 + 4p3 * -2) = 0 6p + -2p2 + (2p + -8p3) = 0 Reorder the terms: 6p + 2p + -2p2 + -8p3 = 0 Combine like terms: 6p + 2p = 8p 8p + -2p2 + -8p3 = 0 Solving 8p + -2p2 + -8p3 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), '2p'. 2p(4 + -1p + -4p2) = 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 '(4 + -1p + -4p2)' equal to zero and attempt to solve: Simplifying 4 + -1p + -4p2 = 0 Solving 4 + -1p + -4p2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1 + 0.25p + p2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.25p + 1 + p2 = 0 + 1 Reorder the terms: -1 + 1 + 0.25p + p2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.25p + p2 = 0 + 1 0.25p + p2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.25p + p2 = 1 The p term is 0.25p. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25p + 0.015625 + p2 = 1 + 0.015625 Reorder the terms: 0.015625 + 0.25p + p2 = 1 + 0.015625 Combine like terms: 1 + 0.015625 = 1.015625 0.015625 + 0.25p + p2 = 1.015625 Factor a perfect square on the left side: (p + 0.125)(p + 0.125) = 1.015625 Calculate the square root of the right side: 1.007782219 Break this problem into two subproblems by setting (p + 0.125) equal to 1.007782219 and -1.007782219.Subproblem 1
p + 0.125 = 1.007782219 Simplifying p + 0.125 = 1.007782219 Reorder the terms: 0.125 + p = 1.007782219 Solving 0.125 + p = 1.007782219 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + p = 1.007782219 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + p = 1.007782219 + -0.125 p = 1.007782219 + -0.125 Combine like terms: 1.007782219 + -0.125 = 0.882782219 p = 0.882782219 Simplifying p = 0.882782219Subproblem 2
p + 0.125 = -1.007782219 Simplifying p + 0.125 = -1.007782219 Reorder the terms: 0.125 + p = -1.007782219 Solving 0.125 + p = -1.007782219 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + p = -1.007782219 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + p = -1.007782219 + -0.125 p = -1.007782219 + -0.125 Combine like terms: -1.007782219 + -0.125 = -1.132782219 p = -1.132782219 Simplifying p = -1.132782219Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.882782219, -1.132782219}Solution
p = {0, 0.882782219, -1.132782219}
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