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Simplifying (p * p * p) + -1(2p * 2p) = (p * p) + 40p Multiply p * p (p2 * p) + -1(2p * 2p) = (p * p) + 40p Multiply p2 * p (p3) + -1(2p * 2p) = (p * p) + 40p p3 + -1(2p * 2p) = (p * p) + 40p Reorder the terms for easier multiplication: p3 + -1(2 * 2p * p) = (p * p) + 40p Multiply 2 * 2 p3 + -1(4p * p) = (p * p) + 40p Multiply p * p p3 + -1(4p2) = (p * p) + 40p Remove parenthesis around (4p2) p3 + -1 * 4p2 = (p * p) + 40p Multiply -1 * 4 p3 + -4p2 = (p * p) + 40p Reorder the terms: -4p2 + p3 = (p * p) + 40p Multiply p * p -4p2 + p3 = (p2) + 40p -4p2 + p3 = p2 + 40p Reorder the terms: -4p2 + p3 = 40p + p2 Solving -4p2 + p3 = 40p + p2 Solving for variable 'p'. Reorder the terms: -40p + -4p2 + -1p2 + p3 = 40p + p2 + -40p + -1p2 Combine like terms: -4p2 + -1p2 = -5p2 -40p + -5p2 + p3 = 40p + p2 + -40p + -1p2 Reorder the terms: -40p + -5p2 + p3 = 40p + -40p + p2 + -1p2 Combine like terms: 40p + -40p = 0 -40p + -5p2 + p3 = 0 + p2 + -1p2 -40p + -5p2 + p3 = p2 + -1p2 Combine like terms: p2 + -1p2 = 0 -40p + -5p2 + p3 = 0 Factor out the Greatest Common Factor (GCF), 'p'. p(-40 + -5p + p2) = 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 '(-40 + -5p + p2)' equal to zero and attempt to solve: Simplifying -40 + -5p + p2 = 0 Solving -40 + -5p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '40' to each side of the equation. -40 + -5p + 40 + p2 = 0 + 40 Reorder the terms: -40 + 40 + -5p + p2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + -5p + p2 = 0 + 40 -5p + p2 = 0 + 40 Combine like terms: 0 + 40 = 40 -5p + p2 = 40 The p term is -5p. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5p + 6.25 + p2 = 40 + 6.25 Reorder the terms: 6.25 + -5p + p2 = 40 + 6.25 Combine like terms: 40 + 6.25 = 46.25 6.25 + -5p + p2 = 46.25 Factor a perfect square on the left side: (p + -2.5)(p + -2.5) = 46.25 Calculate the square root of the right side: 6.800735254 Break this problem into two subproblems by setting (p + -2.5) equal to 6.800735254 and -6.800735254.Subproblem 1
p + -2.5 = 6.800735254 Simplifying p + -2.5 = 6.800735254 Reorder the terms: -2.5 + p = 6.800735254 Solving -2.5 + p = 6.800735254 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + p = 6.800735254 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + p = 6.800735254 + 2.5 p = 6.800735254 + 2.5 Combine like terms: 6.800735254 + 2.5 = 9.300735254 p = 9.300735254 Simplifying p = 9.300735254Subproblem 2
p + -2.5 = -6.800735254 Simplifying p + -2.5 = -6.800735254 Reorder the terms: -2.5 + p = -6.800735254 Solving -2.5 + p = -6.800735254 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + p = -6.800735254 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + p = -6.800735254 + 2.5 p = -6.800735254 + 2.5 Combine like terms: -6.800735254 + 2.5 = -4.300735254 p = -4.300735254 Simplifying p = -4.300735254Solution
The solution to the problem is based on the solutions from the subproblems. p = {9.300735254, -4.300735254}Solution
p = {0, 9.300735254, -4.300735254}
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