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Simplifying -6p(p + 3) = -54 Reorder the terms: -6p(3 + p) = -54 (3 * -6p + p * -6p) = -54 (-18p + -6p2) = -54 Solving -18p + -6p2 = -54 Solving for variable 'p'. Reorder the terms: 54 + -18p + -6p2 = -54 + 54 Combine like terms: -54 + 54 = 0 54 + -18p + -6p2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(9 + -3p + -1p2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(9 + -3p + -1p2)' equal to zero and attempt to solve: Simplifying 9 + -3p + -1p2 = 0 Solving 9 + -3p + -1p2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -9 + 3p + p2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 3p + 9 + p2 = 0 + 9 Reorder the terms: -9 + 9 + 3p + p2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 3p + p2 = 0 + 9 3p + p2 = 0 + 9 Combine like terms: 0 + 9 = 9 3p + p2 = 9 The p term is 3p. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3p + 2.25 + p2 = 9 + 2.25 Reorder the terms: 2.25 + 3p + p2 = 9 + 2.25 Combine like terms: 9 + 2.25 = 11.25 2.25 + 3p + p2 = 11.25 Factor a perfect square on the left side: (p + 1.5)(p + 1.5) = 11.25 Calculate the square root of the right side: 3.354101966 Break this problem into two subproblems by setting (p + 1.5) equal to 3.354101966 and -3.354101966.Subproblem 1
p + 1.5 = 3.354101966 Simplifying p + 1.5 = 3.354101966 Reorder the terms: 1.5 + p = 3.354101966 Solving 1.5 + p = 3.354101966 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = 3.354101966 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = 3.354101966 + -1.5 p = 3.354101966 + -1.5 Combine like terms: 3.354101966 + -1.5 = 1.854101966 p = 1.854101966 Simplifying p = 1.854101966Subproblem 2
p + 1.5 = -3.354101966 Simplifying p + 1.5 = -3.354101966 Reorder the terms: 1.5 + p = -3.354101966 Solving 1.5 + p = -3.354101966 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = -3.354101966 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = -3.354101966 + -1.5 p = -3.354101966 + -1.5 Combine like terms: -3.354101966 + -1.5 = -4.854101966 p = -4.854101966 Simplifying p = -4.854101966Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.854101966, -4.854101966}Solution
p = {1.854101966, -4.854101966}
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