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Simplifying 16p2 + -56p + 43 = 0 Reorder the terms: 43 + -56p + 16p2 = 0 Solving 43 + -56p + 16p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 2.6875 + -3.5p + p2 = 0 Move the constant term to the right: Add '-2.6875' to each side of the equation. 2.6875 + -3.5p + -2.6875 + p2 = 0 + -2.6875 Reorder the terms: 2.6875 + -2.6875 + -3.5p + p2 = 0 + -2.6875 Combine like terms: 2.6875 + -2.6875 = 0.0000 0.0000 + -3.5p + p2 = 0 + -2.6875 -3.5p + p2 = 0 + -2.6875 Combine like terms: 0 + -2.6875 = -2.6875 -3.5p + p2 = -2.6875 The p term is -3.5p. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5p + 3.0625 + p2 = -2.6875 + 3.0625 Reorder the terms: 3.0625 + -3.5p + p2 = -2.6875 + 3.0625 Combine like terms: -2.6875 + 3.0625 = 0.375 3.0625 + -3.5p + p2 = 0.375 Factor a perfect square on the left side: (p + -1.75)(p + -1.75) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (p + -1.75) equal to 0.612372436 and -0.612372436.Subproblem 1
p + -1.75 = 0.612372436 Simplifying p + -1.75 = 0.612372436 Reorder the terms: -1.75 + p = 0.612372436 Solving -1.75 + p = 0.612372436 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + p = 0.612372436 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + p = 0.612372436 + 1.75 p = 0.612372436 + 1.75 Combine like terms: 0.612372436 + 1.75 = 2.362372436 p = 2.362372436 Simplifying p = 2.362372436Subproblem 2
p + -1.75 = -0.612372436 Simplifying p + -1.75 = -0.612372436 Reorder the terms: -1.75 + p = -0.612372436 Solving -1.75 + p = -0.612372436 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + p = -0.612372436 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + p = -0.612372436 + 1.75 p = -0.612372436 + 1.75 Combine like terms: -0.612372436 + 1.75 = 1.137627564 p = 1.137627564 Simplifying p = 1.137627564Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.362372436, 1.137627564}
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