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Simplifying (15 + -0.5p)(p + -2) = 10 Reorder the terms: (15 + -0.5p)(-2 + p) = 10 Multiply (15 + -0.5p) * (-2 + p) (15(-2 + p) + -0.5p * (-2 + p)) = 10 ((-2 * 15 + p * 15) + -0.5p * (-2 + p)) = 10 ((-30 + 15p) + -0.5p * (-2 + p)) = 10 (-30 + 15p + (-2 * -0.5p + p * -0.5p)) = 10 (-30 + 15p + (1p + -0.5p2)) = 10 Combine like terms: 15p + 1p = 16p (-30 + 16p + -0.5p2) = 10 Solving -30 + 16p + -0.5p2 = 10 Solving for variable 'p'. Reorder the terms: -30 + -10 + 16p + -0.5p2 = 10 + -10 Combine like terms: -30 + -10 = -40 -40 + 16p + -0.5p2 = 10 + -10 Combine like terms: 10 + -10 = 0 -40 + 16p + -0.5p2 = 0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. 80 + -32p + p2 = 0 Move the constant term to the right: Add '-80' to each side of the equation. 80 + -32p + -80 + p2 = 0 + -80 Reorder the terms: 80 + -80 + -32p + p2 = 0 + -80 Combine like terms: 80 + -80 = 0 0 + -32p + p2 = 0 + -80 -32p + p2 = 0 + -80 Combine like terms: 0 + -80 = -80 -32p + p2 = -80 The p term is -32p. Take half its coefficient (-16). Square it (256) and add it to both sides. Add '256' to each side of the equation. -32p + 256 + p2 = -80 + 256 Reorder the terms: 256 + -32p + p2 = -80 + 256 Combine like terms: -80 + 256 = 176 256 + -32p + p2 = 176 Factor a perfect square on the left side: (p + -16)(p + -16) = 176 Calculate the square root of the right side: 13.266499161 Break this problem into two subproblems by setting (p + -16) equal to 13.266499161 and -13.266499161.Subproblem 1
p + -16 = 13.266499161 Simplifying p + -16 = 13.266499161 Reorder the terms: -16 + p = 13.266499161 Solving -16 + p = 13.266499161 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + p = 13.266499161 + 16 Combine like terms: -16 + 16 = 0 0 + p = 13.266499161 + 16 p = 13.266499161 + 16 Combine like terms: 13.266499161 + 16 = 29.266499161 p = 29.266499161 Simplifying p = 29.266499161Subproblem 2
p + -16 = -13.266499161 Simplifying p + -16 = -13.266499161 Reorder the terms: -16 + p = -13.266499161 Solving -16 + p = -13.266499161 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + p = -13.266499161 + 16 Combine like terms: -16 + 16 = 0 0 + p = -13.266499161 + 16 p = -13.266499161 + 16 Combine like terms: -13.266499161 + 16 = 2.733500839 p = 2.733500839 Simplifying p = 2.733500839Solution
The solution to the problem is based on the solutions from the subproblems. p = {29.266499161, 2.733500839}
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