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Simplifying 3(2p + 7) = 15p(p + -4) Reorder the terms: 3(7 + 2p) = 15p(p + -4) (7 * 3 + 2p * 3) = 15p(p + -4) (21 + 6p) = 15p(p + -4) Reorder the terms: 21 + 6p = 15p(-4 + p) 21 + 6p = (-4 * 15p + p * 15p) 21 + 6p = (-60p + 15p2) Solving 21 + 6p = -60p + 15p2 Solving for variable 'p'. Combine like terms: 6p + 60p = 66p 21 + 66p + -15p2 = -60p + 15p2 + 60p + -15p2 Reorder the terms: 21 + 66p + -15p2 = -60p + 60p + 15p2 + -15p2 Combine like terms: -60p + 60p = 0 21 + 66p + -15p2 = 0 + 15p2 + -15p2 21 + 66p + -15p2 = 15p2 + -15p2 Combine like terms: 15p2 + -15p2 = 0 21 + 66p + -15p2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(7 + 22p + -5p2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(7 + 22p + -5p2)' equal to zero and attempt to solve: Simplifying 7 + 22p + -5p2 = 0 Solving 7 + 22p + -5p2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -1.4 + -4.4p + p2 = 0 Move the constant term to the right: Add '1.4' to each side of the equation. -1.4 + -4.4p + 1.4 + p2 = 0 + 1.4 Reorder the terms: -1.4 + 1.4 + -4.4p + p2 = 0 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + -4.4p + p2 = 0 + 1.4 -4.4p + p2 = 0 + 1.4 Combine like terms: 0 + 1.4 = 1.4 -4.4p + p2 = 1.4 The p term is -4.4p. Take half its coefficient (-2.2). Square it (4.84) and add it to both sides. Add '4.84' to each side of the equation. -4.4p + 4.84 + p2 = 1.4 + 4.84 Reorder the terms: 4.84 + -4.4p + p2 = 1.4 + 4.84 Combine like terms: 1.4 + 4.84 = 6.24 4.84 + -4.4p + p2 = 6.24 Factor a perfect square on the left side: (p + -2.2)(p + -2.2) = 6.24 Calculate the square root of the right side: 2.497999199 Break this problem into two subproblems by setting (p + -2.2) equal to 2.497999199 and -2.497999199.Subproblem 1
p + -2.2 = 2.497999199 Simplifying p + -2.2 = 2.497999199 Reorder the terms: -2.2 + p = 2.497999199 Solving -2.2 + p = 2.497999199 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.2' to each side of the equation. -2.2 + 2.2 + p = 2.497999199 + 2.2 Combine like terms: -2.2 + 2.2 = 0.0 0.0 + p = 2.497999199 + 2.2 p = 2.497999199 + 2.2 Combine like terms: 2.497999199 + 2.2 = 4.697999199 p = 4.697999199 Simplifying p = 4.697999199Subproblem 2
p + -2.2 = -2.497999199 Simplifying p + -2.2 = -2.497999199 Reorder the terms: -2.2 + p = -2.497999199 Solving -2.2 + p = -2.497999199 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.2' to each side of the equation. -2.2 + 2.2 + p = -2.497999199 + 2.2 Combine like terms: -2.2 + 2.2 = 0.0 0.0 + p = -2.497999199 + 2.2 p = -2.497999199 + 2.2 Combine like terms: -2.497999199 + 2.2 = -0.297999199 p = -0.297999199 Simplifying p = -0.297999199Solution
The solution to the problem is based on the solutions from the subproblems. p = {4.697999199, -0.297999199}Solution
p = {4.697999199, -0.297999199}
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