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Simplifying 7p2 + -7p + -24 = -5 Reorder the terms: -24 + -7p + 7p2 = -5 Solving -24 + -7p + 7p2 = -5 Solving for variable 'p'. Reorder the terms: -24 + 5 + -7p + 7p2 = -5 + 5 Combine like terms: -24 + 5 = -19 -19 + -7p + 7p2 = -5 + 5 Combine like terms: -5 + 5 = 0 -19 + -7p + 7p2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.714285714 + -1p + p2 = 0 Move the constant term to the right: Add '2.714285714' to each side of the equation. -2.714285714 + -1p + 2.714285714 + p2 = 0 + 2.714285714 Reorder the terms: -2.714285714 + 2.714285714 + -1p + p2 = 0 + 2.714285714 Combine like terms: -2.714285714 + 2.714285714 = 0.000000000 0.000000000 + -1p + p2 = 0 + 2.714285714 -1p + p2 = 0 + 2.714285714 Combine like terms: 0 + 2.714285714 = 2.714285714 -1p + p2 = 2.714285714 The p term is -1p. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1p + 0.25 + p2 = 2.714285714 + 0.25 Reorder the terms: 0.25 + -1p + p2 = 2.714285714 + 0.25 Combine like terms: 2.714285714 + 0.25 = 2.964285714 0.25 + -1p + p2 = 2.964285714 Factor a perfect square on the left side: (p + -0.5)(p + -0.5) = 2.964285714 Calculate the square root of the right side: 1.721710113 Break this problem into two subproblems by setting (p + -0.5) equal to 1.721710113 and -1.721710113.Subproblem 1
p + -0.5 = 1.721710113 Simplifying p + -0.5 = 1.721710113 Reorder the terms: -0.5 + p = 1.721710113 Solving -0.5 + p = 1.721710113 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + p = 1.721710113 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = 1.721710113 + 0.5 p = 1.721710113 + 0.5 Combine like terms: 1.721710113 + 0.5 = 2.221710113 p = 2.221710113 Simplifying p = 2.221710113Subproblem 2
p + -0.5 = -1.721710113 Simplifying p + -0.5 = -1.721710113 Reorder the terms: -0.5 + p = -1.721710113 Solving -0.5 + p = -1.721710113 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + p = -1.721710113 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = -1.721710113 + 0.5 p = -1.721710113 + 0.5 Combine like terms: -1.721710113 + 0.5 = -1.221710113 p = -1.221710113 Simplifying p = -1.221710113Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.221710113, -1.221710113}
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