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Simplifying 3(5p + 9) * 12p = 51 Reorder the terms: 3(9 + 5p) * 12p = 51 Reorder the terms for easier multiplication: 3 * 12p(9 + 5p) = 51 Multiply 3 * 12 36p(9 + 5p) = 51 (9 * 36p + 5p * 36p) = 51 (324p + 180p2) = 51 Solving 324p + 180p2 = 51 Solving for variable 'p'. Reorder the terms: -51 + 324p + 180p2 = 51 + -51 Combine like terms: 51 + -51 = 0 -51 + 324p + 180p2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-17 + 108p + 60p2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-17 + 108p + 60p2)' equal to zero and attempt to solve: Simplifying -17 + 108p + 60p2 = 0 Solving -17 + 108p + 60p2 = 0 Begin completing the square. Divide all terms by 60 the coefficient of the squared term: Divide each side by '60'. -0.2833333333 + 1.8p + p2 = 0.0 Move the constant term to the right: Add '0.2833333333' to each side of the equation. -0.2833333333 + 1.8p + 0.2833333333 + p2 = 0.0 + 0.2833333333 Reorder the terms: -0.2833333333 + 0.2833333333 + 1.8p + p2 = 0.0 + 0.2833333333 Combine like terms: -0.2833333333 + 0.2833333333 = 0.0000000000 0.0000000000 + 1.8p + p2 = 0.0 + 0.2833333333 1.8p + p2 = 0.0 + 0.2833333333 Combine like terms: 0.0 + 0.2833333333 = 0.2833333333 1.8p + p2 = 0.2833333333 The p term is 1.8p. Take half its coefficient (0.9). Square it (0.81) and add it to both sides. Add '0.81' to each side of the equation. 1.8p + 0.81 + p2 = 0.2833333333 + 0.81 Reorder the terms: 0.81 + 1.8p + p2 = 0.2833333333 + 0.81 Combine like terms: 0.2833333333 + 0.81 = 1.0933333333 0.81 + 1.8p + p2 = 1.0933333333 Factor a perfect square on the left side: (p + 0.9)(p + 0.9) = 1.0933333333 Calculate the square root of the right side: 1.045625809 Break this problem into two subproblems by setting (p + 0.9) equal to 1.045625809 and -1.045625809.Subproblem 1
p + 0.9 = 1.045625809 Simplifying p + 0.9 = 1.045625809 Reorder the terms: 0.9 + p = 1.045625809 Solving 0.9 + p = 1.045625809 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + p = 1.045625809 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + p = 1.045625809 + -0.9 p = 1.045625809 + -0.9 Combine like terms: 1.045625809 + -0.9 = 0.145625809 p = 0.145625809 Simplifying p = 0.145625809Subproblem 2
p + 0.9 = -1.045625809 Simplifying p + 0.9 = -1.045625809 Reorder the terms: 0.9 + p = -1.045625809 Solving 0.9 + p = -1.045625809 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + p = -1.045625809 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + p = -1.045625809 + -0.9 p = -1.045625809 + -0.9 Combine like terms: -1.045625809 + -0.9 = -1.945625809 p = -1.945625809 Simplifying p = -1.945625809Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.145625809, -1.945625809}Solution
p = {0.145625809, -1.945625809}
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