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Simplifying -10p + -8 = -8p(6p + -5) + 10(2 + 4p) Reorder the terms: -8 + -10p = -8p(6p + -5) + 10(2 + 4p) Reorder the terms: -8 + -10p = -8p(-5 + 6p) + 10(2 + 4p) -8 + -10p = (-5 * -8p + 6p * -8p) + 10(2 + 4p) -8 + -10p = (40p + -48p2) + 10(2 + 4p) -8 + -10p = 40p + -48p2 + (2 * 10 + 4p * 10) -8 + -10p = 40p + -48p2 + (20 + 40p) Reorder the terms: -8 + -10p = 20 + 40p + 40p + -48p2 Combine like terms: 40p + 40p = 80p -8 + -10p = 20 + 80p + -48p2 Solving -8 + -10p = 20 + 80p + -48p2 Solving for variable 'p'. Reorder the terms: -8 + -20 + -10p + -80p + 48p2 = 20 + 80p + -48p2 + -20 + -80p + 48p2 Combine like terms: -8 + -20 = -28 -28 + -10p + -80p + 48p2 = 20 + 80p + -48p2 + -20 + -80p + 48p2 Combine like terms: -10p + -80p = -90p -28 + -90p + 48p2 = 20 + 80p + -48p2 + -20 + -80p + 48p2 Reorder the terms: -28 + -90p + 48p2 = 20 + -20 + 80p + -80p + -48p2 + 48p2 Combine like terms: 20 + -20 = 0 -28 + -90p + 48p2 = 0 + 80p + -80p + -48p2 + 48p2 -28 + -90p + 48p2 = 80p + -80p + -48p2 + 48p2 Combine like terms: 80p + -80p = 0 -28 + -90p + 48p2 = 0 + -48p2 + 48p2 -28 + -90p + 48p2 = -48p2 + 48p2 Combine like terms: -48p2 + 48p2 = 0 -28 + -90p + 48p2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-14 + -45p + 24p2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-14 + -45p + 24p2)' equal to zero and attempt to solve: Simplifying -14 + -45p + 24p2 = 0 Solving -14 + -45p + 24p2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -0.5833333333 + -1.875p + p2 = 0 Move the constant term to the right: Add '0.5833333333' to each side of the equation. -0.5833333333 + -1.875p + 0.5833333333 + p2 = 0 + 0.5833333333 Reorder the terms: -0.5833333333 + 0.5833333333 + -1.875p + p2 = 0 + 0.5833333333 Combine like terms: -0.5833333333 + 0.5833333333 = 0.0000000000 0.0000000000 + -1.875p + p2 = 0 + 0.5833333333 -1.875p + p2 = 0 + 0.5833333333 Combine like terms: 0 + 0.5833333333 = 0.5833333333 -1.875p + p2 = 0.5833333333 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 = 0.5833333333 + 0.87890625 Reorder the terms: 0.87890625 + -1.875p + p2 = 0.5833333333 + 0.87890625 Combine like terms: 0.5833333333 + 0.87890625 = 1.4622395833 0.87890625 + -1.875p + p2 = 1.4622395833 Factor a perfect square on the left side: (p + -0.9375)(p + -0.9375) = 1.4622395833 Calculate the square root of the right side: 1.209230988 Break this problem into two subproblems by setting (p + -0.9375) equal to 1.209230988 and -1.209230988.Subproblem 1
p + -0.9375 = 1.209230988 Simplifying p + -0.9375 = 1.209230988 Reorder the terms: -0.9375 + p = 1.209230988 Solving -0.9375 + p = 1.209230988 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 = 1.209230988 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + p = 1.209230988 + 0.9375 p = 1.209230988 + 0.9375 Combine like terms: 1.209230988 + 0.9375 = 2.146730988 p = 2.146730988 Simplifying p = 2.146730988Subproblem 2
p + -0.9375 = -1.209230988 Simplifying p + -0.9375 = -1.209230988 Reorder the terms: -0.9375 + p = -1.209230988 Solving -0.9375 + p = -1.209230988 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 = -1.209230988 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + p = -1.209230988 + 0.9375 p = -1.209230988 + 0.9375 Combine like terms: -1.209230988 + 0.9375 = -0.271730988 p = -0.271730988 Simplifying p = -0.271730988Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.146730988, -0.271730988}Solution
p = {2.146730988, -0.271730988}
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