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Simplifying 3.50p + 1.50p(32 + -1p) = 2.75 * 32 3.50p + (32 * 1.50p + -1p * 1.50p) = 2.75 * 32 3.50p + (48p + -1.5p2) = 2.75 * 32 Combine like terms: 3.50p + 48p = 51.5p 51.5p + -1.5p2 = 2.75 * 32 Multiply 2.75 * 32 51.5p + -1.5p2 = 88 Solving 51.5p + -1.5p2 = 88 Solving for variable 'p'. Reorder the terms: -88 + 51.5p + -1.5p2 = 88 + -88 Combine like terms: 88 + -88 = 0 -88 + 51.5p + -1.5p2 = 0 Begin completing the square. Divide all terms by -1.5 the coefficient of the squared term: Divide each side by '-1.5'. 58.66666667 + -34.33333333p + p2 = 0 Move the constant term to the right: Add '-58.66666667' to each side of the equation. 58.66666667 + -34.33333333p + -58.66666667 + p2 = 0 + -58.66666667 Reorder the terms: 58.66666667 + -58.66666667 + -34.33333333p + p2 = 0 + -58.66666667 Combine like terms: 58.66666667 + -58.66666667 = 0.00000000 0.00000000 + -34.33333333p + p2 = 0 + -58.66666667 -34.33333333p + p2 = 0 + -58.66666667 Combine like terms: 0 + -58.66666667 = -58.66666667 -34.33333333p + p2 = -58.66666667 The p term is -34.33333333p. Take half its coefficient (-17.16666667). Square it (294.6944446) and add it to both sides. Add '294.6944446' to each side of the equation. -34.33333333p + 294.6944446 + p2 = -58.66666667 + 294.6944446 Reorder the terms: 294.6944446 + -34.33333333p + p2 = -58.66666667 + 294.6944446 Combine like terms: -58.66666667 + 294.6944446 = 236.02777793 294.6944446 + -34.33333333p + p2 = 236.02777793 Factor a perfect square on the left side: (p + -17.16666667)(p + -17.16666667) = 236.02777793 Calculate the square root of the right side: 15.363195564 Break this problem into two subproblems by setting (p + -17.16666667) equal to 15.363195564 and -15.363195564.Subproblem 1
p + -17.16666667 = 15.363195564 Simplifying p + -17.16666667 = 15.363195564 Reorder the terms: -17.16666667 + p = 15.363195564 Solving -17.16666667 + p = 15.363195564 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '17.16666667' to each side of the equation. -17.16666667 + 17.16666667 + p = 15.363195564 + 17.16666667 Combine like terms: -17.16666667 + 17.16666667 = 0.00000000 0.00000000 + p = 15.363195564 + 17.16666667 p = 15.363195564 + 17.16666667 Combine like terms: 15.363195564 + 17.16666667 = 32.529862234 p = 32.529862234 Simplifying p = 32.529862234Subproblem 2
p + -17.16666667 = -15.363195564 Simplifying p + -17.16666667 = -15.363195564 Reorder the terms: -17.16666667 + p = -15.363195564 Solving -17.16666667 + p = -15.363195564 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '17.16666667' to each side of the equation. -17.16666667 + 17.16666667 + p = -15.363195564 + 17.16666667 Combine like terms: -17.16666667 + 17.16666667 = 0.00000000 0.00000000 + p = -15.363195564 + 17.16666667 p = -15.363195564 + 17.16666667 Combine like terms: -15.363195564 + 17.16666667 = 1.803471106 p = 1.803471106 Simplifying p = 1.803471106Solution
The solution to the problem is based on the solutions from the subproblems. p = {32.529862234, 1.803471106}
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