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Simplifying 0 = p2 + -9p + -32 Reorder the terms: 0 = -32 + -9p + p2 Solving 0 = -32 + -9p + p2 Solving for variable 'p'. Combine like terms: 0 + 32 = 32 32 + 9p + -1p2 = -32 + -9p + p2 + 32 + 9p + -1p2 Reorder the terms: 32 + 9p + -1p2 = -32 + 32 + -9p + 9p + p2 + -1p2 Combine like terms: -32 + 32 = 0 32 + 9p + -1p2 = 0 + -9p + 9p + p2 + -1p2 32 + 9p + -1p2 = -9p + 9p + p2 + -1p2 Combine like terms: -9p + 9p = 0 32 + 9p + -1p2 = 0 + p2 + -1p2 32 + 9p + -1p2 = p2 + -1p2 Combine like terms: p2 + -1p2 = 0 32 + 9p + -1p2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -32 + -9p + p2 = 0 Move the constant term to the right: Add '32' to each side of the equation. -32 + -9p + 32 + p2 = 0 + 32 Reorder the terms: -32 + 32 + -9p + p2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + -9p + p2 = 0 + 32 -9p + p2 = 0 + 32 Combine like terms: 0 + 32 = 32 -9p + p2 = 32 The p term is -9p. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9p + 20.25 + p2 = 32 + 20.25 Reorder the terms: 20.25 + -9p + p2 = 32 + 20.25 Combine like terms: 32 + 20.25 = 52.25 20.25 + -9p + p2 = 52.25 Factor a perfect square on the left side: (p + -4.5)(p + -4.5) = 52.25 Calculate the square root of the right side: 7.228416147 Break this problem into two subproblems by setting (p + -4.5) equal to 7.228416147 and -7.228416147.Subproblem 1
p + -4.5 = 7.228416147 Simplifying p + -4.5 = 7.228416147 Reorder the terms: -4.5 + p = 7.228416147 Solving -4.5 + p = 7.228416147 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + p = 7.228416147 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + p = 7.228416147 + 4.5 p = 7.228416147 + 4.5 Combine like terms: 7.228416147 + 4.5 = 11.728416147 p = 11.728416147 Simplifying p = 11.728416147Subproblem 2
p + -4.5 = -7.228416147 Simplifying p + -4.5 = -7.228416147 Reorder the terms: -4.5 + p = -7.228416147 Solving -4.5 + p = -7.228416147 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + p = -7.228416147 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + p = -7.228416147 + 4.5 p = -7.228416147 + 4.5 Combine like terms: -7.228416147 + 4.5 = -2.728416147 p = -2.728416147 Simplifying p = -2.728416147Solution
The solution to the problem is based on the solutions from the subproblems. p = {11.728416147, -2.728416147}
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