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Simplifying 16n2 + -56n = -47 Reorder the terms: -56n + 16n2 = -47 Solving -56n + 16n2 = -47 Solving for variable 'n'. Reorder the terms: 47 + -56n + 16n2 = -47 + 47 Combine like terms: -47 + 47 = 0 47 + -56n + 16n2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 2.9375 + -3.5n + n2 = 0 Move the constant term to the right: Add '-2.9375' to each side of the equation. 2.9375 + -3.5n + -2.9375 + n2 = 0 + -2.9375 Reorder the terms: 2.9375 + -2.9375 + -3.5n + n2 = 0 + -2.9375 Combine like terms: 2.9375 + -2.9375 = 0.0000 0.0000 + -3.5n + n2 = 0 + -2.9375 -3.5n + n2 = 0 + -2.9375 Combine like terms: 0 + -2.9375 = -2.9375 -3.5n + n2 = -2.9375 The n term is -3.5n. 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. -3.5n + 3.0625 + n2 = -2.9375 + 3.0625 Reorder the terms: 3.0625 + -3.5n + n2 = -2.9375 + 3.0625 Combine like terms: -2.9375 + 3.0625 = 0.125 3.0625 + -3.5n + n2 = 0.125 Factor a perfect square on the left side: (n + -1.75)(n + -1.75) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (n + -1.75) equal to 0.353553391 and -0.353553391.Subproblem 1
n + -1.75 = 0.353553391 Simplifying n + -1.75 = 0.353553391 Reorder the terms: -1.75 + n = 0.353553391 Solving -1.75 + n = 0.353553391 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = 0.353553391 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = 0.353553391 + 1.75 n = 0.353553391 + 1.75 Combine like terms: 0.353553391 + 1.75 = 2.103553391 n = 2.103553391 Simplifying n = 2.103553391Subproblem 2
n + -1.75 = -0.353553391 Simplifying n + -1.75 = -0.353553391 Reorder the terms: -1.75 + n = -0.353553391 Solving -1.75 + n = -0.353553391 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = -0.353553391 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = -0.353553391 + 1.75 n = -0.353553391 + 1.75 Combine like terms: -0.353553391 + 1.75 = 1.396446609 n = 1.396446609 Simplifying n = 1.396446609Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.103553391, 1.396446609}
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