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Simplifying 5p3 + p2 + -10p = 0 Reorder the terms: -10p + p2 + 5p3 = 0 Solving -10p + p2 + 5p3 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), 'p'. p(-10 + p + 5p2) = 0Subproblem 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 '(-10 + p + 5p2)' equal to zero and attempt to solve: Simplifying -10 + p + 5p2 = 0 Solving -10 + p + 5p2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2 + 0.2p + p2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.2p + 2 + p2 = 0 + 2 Reorder the terms: -2 + 2 + 0.2p + p2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.2p + p2 = 0 + 2 0.2p + p2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.2p + p2 = 2 The p term is p. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.2p + 0.25 + p2 = 2 + 0.25 Reorder the terms: 0.25 + 0.2p + p2 = 2 + 0.25 Combine like terms: 2 + 0.25 = 2.25 0.25 + 0.2p + p2 = 2.25 Factor a perfect square on the left side: (p + 0.5)(p + 0.5) = 2.25 Calculate the square root of the right side: 1.5 Break this problem into two subproblems by setting (p + 0.5) equal to 1.5 and -1.5.Subproblem 1
p + 0.5 = 1.5 Simplifying p + 0.5 = 1.5 Reorder the terms: 0.5 + p = 1.5 Solving 0.5 + p = 1.5 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + p = 1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = 1.5 + -0.5 p = 1.5 + -0.5 Combine like terms: 1.5 + -0.5 = 1 p = 1 Simplifying p = 1Subproblem 2
p + 0.5 = -1.5 Simplifying p + 0.5 = -1.5 Reorder the terms: 0.5 + p = -1.5 Solving 0.5 + p = -1.5 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + p = -1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = -1.5 + -0.5 p = -1.5 + -0.5 Combine like terms: -1.5 + -0.5 = -2 p = -2 Simplifying p = -2Solution
The solution to the problem is based on the solutions from the subproblems. p = {1, -2}Solution
p = {0, 1, -2}
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