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Simplifying 5 + 3(-11p + 3) = -4p(8p + -5) Reorder the terms: 5 + 3(3 + -11p) = -4p(8p + -5) 5 + (3 * 3 + -11p * 3) = -4p(8p + -5) 5 + (9 + -33p) = -4p(8p + -5) Combine like terms: 5 + 9 = 14 14 + -33p = -4p(8p + -5) Reorder the terms: 14 + -33p = -4p(-5 + 8p) 14 + -33p = (-5 * -4p + 8p * -4p) 14 + -33p = (20p + -32p2) Solving 14 + -33p = 20p + -32p2 Solving for variable 'p'. Combine like terms: -33p + -20p = -53p 14 + -53p + 32p2 = 20p + -32p2 + -20p + 32p2 Reorder the terms: 14 + -53p + 32p2 = 20p + -20p + -32p2 + 32p2 Combine like terms: 20p + -20p = 0 14 + -53p + 32p2 = 0 + -32p2 + 32p2 14 + -53p + 32p2 = -32p2 + 32p2 Combine like terms: -32p2 + 32p2 = 0 14 + -53p + 32p2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. 0.4375 + -1.65625p + p2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + -1.65625p + -0.4375 + p2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + -1.65625p + p2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + -1.65625p + p2 = 0 + -0.4375 -1.65625p + p2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 -1.65625p + p2 = -0.4375 The p term is -1.65625p. Take half its coefficient (-0.828125). Square it (0.6857910156) and add it to both sides. Add '0.6857910156' to each side of the equation. -1.65625p + 0.6857910156 + p2 = -0.4375 + 0.6857910156 Reorder the terms: 0.6857910156 + -1.65625p + p2 = -0.4375 + 0.6857910156 Combine like terms: -0.4375 + 0.6857910156 = 0.2482910156 0.6857910156 + -1.65625p + p2 = 0.2482910156 Factor a perfect square on the left side: (p + -0.828125)(p + -0.828125) = 0.2482910156 Calculate the square root of the right side: 0.498288085 Break this problem into two subproblems by setting (p + -0.828125) equal to 0.498288085 and -0.498288085.Subproblem 1
p + -0.828125 = 0.498288085 Simplifying p + -0.828125 = 0.498288085 Reorder the terms: -0.828125 + p = 0.498288085 Solving -0.828125 + p = 0.498288085 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.828125' to each side of the equation. -0.828125 + 0.828125 + p = 0.498288085 + 0.828125 Combine like terms: -0.828125 + 0.828125 = 0.000000 0.000000 + p = 0.498288085 + 0.828125 p = 0.498288085 + 0.828125 Combine like terms: 0.498288085 + 0.828125 = 1.326413085 p = 1.326413085 Simplifying p = 1.326413085Subproblem 2
p + -0.828125 = -0.498288085 Simplifying p + -0.828125 = -0.498288085 Reorder the terms: -0.828125 + p = -0.498288085 Solving -0.828125 + p = -0.498288085 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.828125' to each side of the equation. -0.828125 + 0.828125 + p = -0.498288085 + 0.828125 Combine like terms: -0.828125 + 0.828125 = 0.000000 0.000000 + p = -0.498288085 + 0.828125 p = -0.498288085 + 0.828125 Combine like terms: -0.498288085 + 0.828125 = 0.329836915 p = 0.329836915 Simplifying p = 0.329836915Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.326413085, 0.329836915}
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