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Simplifying 4p2 + -20p + 23 = 0 Reorder the terms: 23 + -20p + 4p2 = 0 Solving 23 + -20p + 4p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 5.75 + -5p + p2 = 0 Move the constant term to the right: Add '-5.75' to each side of the equation. 5.75 + -5p + -5.75 + p2 = 0 + -5.75 Reorder the terms: 5.75 + -5.75 + -5p + p2 = 0 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + -5p + p2 = 0 + -5.75 -5p + p2 = 0 + -5.75 Combine like terms: 0 + -5.75 = -5.75 -5p + p2 = -5.75 The p term is -5p. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5p + 6.25 + p2 = -5.75 + 6.25 Reorder the terms: 6.25 + -5p + p2 = -5.75 + 6.25 Combine like terms: -5.75 + 6.25 = 0.5 6.25 + -5p + p2 = 0.5 Factor a perfect square on the left side: (p + -2.5)(p + -2.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (p + -2.5) equal to 0.707106781 and -0.707106781.Subproblem 1
p + -2.5 = 0.707106781 Simplifying p + -2.5 = 0.707106781 Reorder the terms: -2.5 + p = 0.707106781 Solving -2.5 + p = 0.707106781 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + p = 0.707106781 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + p = 0.707106781 + 2.5 p = 0.707106781 + 2.5 Combine like terms: 0.707106781 + 2.5 = 3.207106781 p = 3.207106781 Simplifying p = 3.207106781Subproblem 2
p + -2.5 = -0.707106781 Simplifying p + -2.5 = -0.707106781 Reorder the terms: -2.5 + p = -0.707106781 Solving -2.5 + p = -0.707106781 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + p = -0.707106781 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + p = -0.707106781 + 2.5 p = -0.707106781 + 2.5 Combine like terms: -0.707106781 + 2.5 = 1.792893219 p = 1.792893219 Simplifying p = 1.792893219Solution
The solution to the problem is based on the solutions from the subproblems. p = {3.207106781, 1.792893219}
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