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Simplifying 12p(2p + -3) = 99 + 9p Reorder the terms: 12p(-3 + 2p) = 99 + 9p (-3 * 12p + 2p * 12p) = 99 + 9p (-36p + 24p2) = 99 + 9p Solving -36p + 24p2 = 99 + 9p Solving for variable 'p'. Reorder the terms: -99 + -36p + -9p + 24p2 = 99 + 9p + -99 + -9p Combine like terms: -36p + -9p = -45p -99 + -45p + 24p2 = 99 + 9p + -99 + -9p Reorder the terms: -99 + -45p + 24p2 = 99 + -99 + 9p + -9p Combine like terms: 99 + -99 = 0 -99 + -45p + 24p2 = 0 + 9p + -9p -99 + -45p + 24p2 = 9p + -9p Combine like terms: 9p + -9p = 0 -99 + -45p + 24p2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-33 + -15p + 8p2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-33 + -15p + 8p2)' equal to zero and attempt to solve: Simplifying -33 + -15p + 8p2 = 0 Solving -33 + -15p + 8p2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -4.125 + -1.875p + p2 = 0 Move the constant term to the right: Add '4.125' to each side of the equation. -4.125 + -1.875p + 4.125 + p2 = 0 + 4.125 Reorder the terms: -4.125 + 4.125 + -1.875p + p2 = 0 + 4.125 Combine like terms: -4.125 + 4.125 = 0.000 0.000 + -1.875p + p2 = 0 + 4.125 -1.875p + p2 = 0 + 4.125 Combine like terms: 0 + 4.125 = 4.125 -1.875p + p2 = 4.125 The p term is -1.875p. Take half its coefficient (-0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. -1.875p + 0.87890625 + p2 = 4.125 + 0.87890625 Reorder the terms: 0.87890625 + -1.875p + p2 = 4.125 + 0.87890625 Combine like terms: 4.125 + 0.87890625 = 5.00390625 0.87890625 + -1.875p + p2 = 5.00390625 Factor a perfect square on the left side: (p + -0.9375)(p + -0.9375) = 5.00390625 Calculate the square root of the right side: 2.236941271 Break this problem into two subproblems by setting (p + -0.9375) equal to 2.236941271 and -2.236941271.Subproblem 1
p + -0.9375 = 2.236941271 Simplifying p + -0.9375 = 2.236941271 Reorder the terms: -0.9375 + p = 2.236941271 Solving -0.9375 + p = 2.236941271 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + p = 2.236941271 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + p = 2.236941271 + 0.9375 p = 2.236941271 + 0.9375 Combine like terms: 2.236941271 + 0.9375 = 3.174441271 p = 3.174441271 Simplifying p = 3.174441271Subproblem 2
p + -0.9375 = -2.236941271 Simplifying p + -0.9375 = -2.236941271 Reorder the terms: -0.9375 + p = -2.236941271 Solving -0.9375 + p = -2.236941271 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + p = -2.236941271 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + p = -2.236941271 + 0.9375 p = -2.236941271 + 0.9375 Combine like terms: -2.236941271 + 0.9375 = -1.299441271 p = -1.299441271 Simplifying p = -1.299441271Solution
The solution to the problem is based on the solutions from the subproblems. p = {3.174441271, -1.299441271}Solution
p = {3.174441271, -1.299441271}
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