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Simplifying 23p + -24 = p2 Reorder the terms: -24 + 23p = p2 Solving -24 + 23p = p2 Solving for variable 'p'. Combine like terms: p2 + -1p2 = 0 -24 + 23p + -1p2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 24 + -23p + p2 = 0 Move the constant term to the right: Add '-24' to each side of the equation. 24 + -23p + -24 + p2 = 0 + -24 Reorder the terms: 24 + -24 + -23p + p2 = 0 + -24 Combine like terms: 24 + -24 = 0 0 + -23p + p2 = 0 + -24 -23p + p2 = 0 + -24 Combine like terms: 0 + -24 = -24 -23p + p2 = -24 The p term is -23p. Take half its coefficient (-11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. -23p + 132.25 + p2 = -24 + 132.25 Reorder the terms: 132.25 + -23p + p2 = -24 + 132.25 Combine like terms: -24 + 132.25 = 108.25 132.25 + -23p + p2 = 108.25 Factor a perfect square on the left side: (p + -11.5)(p + -11.5) = 108.25 Calculate the square root of the right side: 10.404326023 Break this problem into two subproblems by setting (p + -11.5) equal to 10.404326023 and -10.404326023.Subproblem 1
p + -11.5 = 10.404326023 Simplifying p + -11.5 = 10.404326023 Reorder the terms: -11.5 + p = 10.404326023 Solving -11.5 + p = 10.404326023 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + p = 10.404326023 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + p = 10.404326023 + 11.5 p = 10.404326023 + 11.5 Combine like terms: 10.404326023 + 11.5 = 21.904326023 p = 21.904326023 Simplifying p = 21.904326023Subproblem 2
p + -11.5 = -10.404326023 Simplifying p + -11.5 = -10.404326023 Reorder the terms: -11.5 + p = -10.404326023 Solving -11.5 + p = -10.404326023 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + p = -10.404326023 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + p = -10.404326023 + 11.5 p = -10.404326023 + 11.5 Combine like terms: -10.404326023 + 11.5 = 1.095673977 p = 1.095673977 Simplifying p = 1.095673977Solution
The solution to the problem is based on the solutions from the subproblems. p = {21.904326023, 1.095673977}
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