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Simplifying 6p + -1(5p + 5) = -8 + -2p(p + 12) Reorder the terms: 6p + -1(5 + 5p) = -8 + -2p(p + 12) 6p + (5 * -1 + 5p * -1) = -8 + -2p(p + 12) 6p + (-5 + -5p) = -8 + -2p(p + 12) Reorder the terms: -5 + 6p + -5p = -8 + -2p(p + 12) Combine like terms: 6p + -5p = 1p -5 + 1p = -8 + -2p(p + 12) Reorder the terms: -5 + 1p = -8 + -2p(12 + p) -5 + 1p = -8 + (12 * -2p + p * -2p) -5 + 1p = -8 + (-24p + -2p2) Solving -5 + 1p = -8 + -24p + -2p2 Solving for variable 'p'. Reorder the terms: -5 + 8 + 1p + 24p + 2p2 = -8 + -24p + -2p2 + 8 + 24p + 2p2 Combine like terms: -5 + 8 = 3 3 + 1p + 24p + 2p2 = -8 + -24p + -2p2 + 8 + 24p + 2p2 Combine like terms: 1p + 24p = 25p 3 + 25p + 2p2 = -8 + -24p + -2p2 + 8 + 24p + 2p2 Reorder the terms: 3 + 25p + 2p2 = -8 + 8 + -24p + 24p + -2p2 + 2p2 Combine like terms: -8 + 8 = 0 3 + 25p + 2p2 = 0 + -24p + 24p + -2p2 + 2p2 3 + 25p + 2p2 = -24p + 24p + -2p2 + 2p2 Combine like terms: -24p + 24p = 0 3 + 25p + 2p2 = 0 + -2p2 + 2p2 3 + 25p + 2p2 = -2p2 + 2p2 Combine like terms: -2p2 + 2p2 = 0 3 + 25p + 2p2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + 12.5p + p2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 12.5p + -1.5 + p2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 12.5p + p2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 12.5p + p2 = 0 + -1.5 12.5p + p2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 12.5p + p2 = -1.5 The p term is 12.5p. Take half its coefficient (6.25). Square it (39.0625) and add it to both sides. Add '39.0625' to each side of the equation. 12.5p + 39.0625 + p2 = -1.5 + 39.0625 Reorder the terms: 39.0625 + 12.5p + p2 = -1.5 + 39.0625 Combine like terms: -1.5 + 39.0625 = 37.5625 39.0625 + 12.5p + p2 = 37.5625 Factor a perfect square on the left side: (p + 6.25)(p + 6.25) = 37.5625 Calculate the square root of the right side: 6.128825336 Break this problem into two subproblems by setting (p + 6.25) equal to 6.128825336 and -6.128825336.Subproblem 1
p + 6.25 = 6.128825336 Simplifying p + 6.25 = 6.128825336 Reorder the terms: 6.25 + p = 6.128825336 Solving 6.25 + p = 6.128825336 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-6.25' to each side of the equation. 6.25 + -6.25 + p = 6.128825336 + -6.25 Combine like terms: 6.25 + -6.25 = 0.00 0.00 + p = 6.128825336 + -6.25 p = 6.128825336 + -6.25 Combine like terms: 6.128825336 + -6.25 = -0.121174664 p = -0.121174664 Simplifying p = -0.121174664Subproblem 2
p + 6.25 = -6.128825336 Simplifying p + 6.25 = -6.128825336 Reorder the terms: 6.25 + p = -6.128825336 Solving 6.25 + p = -6.128825336 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-6.25' to each side of the equation. 6.25 + -6.25 + p = -6.128825336 + -6.25 Combine like terms: 6.25 + -6.25 = 0.00 0.00 + p = -6.128825336 + -6.25 p = -6.128825336 + -6.25 Combine like terms: -6.128825336 + -6.25 = -12.378825336 p = -12.378825336 Simplifying p = -12.378825336Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.121174664, -12.378825336}
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