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Simplifying 16n2 + -40n = -23 Reorder the terms: -40n + 16n2 = -23 Solving -40n + 16n2 = -23 Solving for variable 'n'. Reorder the terms: 23 + -40n + 16n2 = -23 + 23 Combine like terms: -23 + 23 = 0 23 + -40n + 16n2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.4375 + -2.5n + n2 = 0 Move the constant term to the right: Add '-1.4375' to each side of the equation. 1.4375 + -2.5n + -1.4375 + n2 = 0 + -1.4375 Reorder the terms: 1.4375 + -1.4375 + -2.5n + n2 = 0 + -1.4375 Combine like terms: 1.4375 + -1.4375 = 0.0000 0.0000 + -2.5n + n2 = 0 + -1.4375 -2.5n + n2 = 0 + -1.4375 Combine like terms: 0 + -1.4375 = -1.4375 -2.5n + n2 = -1.4375 The n term is -2.5n. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5n + 1.5625 + n2 = -1.4375 + 1.5625 Reorder the terms: 1.5625 + -2.5n + n2 = -1.4375 + 1.5625 Combine like terms: -1.4375 + 1.5625 = 0.125 1.5625 + -2.5n + n2 = 0.125 Factor a perfect square on the left side: (n + -1.25)(n + -1.25) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (n + -1.25) equal to 0.353553391 and -0.353553391.Subproblem 1
n + -1.25 = 0.353553391 Simplifying n + -1.25 = 0.353553391 Reorder the terms: -1.25 + n = 0.353553391 Solving -1.25 + n = 0.353553391 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + n = 0.353553391 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + n = 0.353553391 + 1.25 n = 0.353553391 + 1.25 Combine like terms: 0.353553391 + 1.25 = 1.603553391 n = 1.603553391 Simplifying n = 1.603553391Subproblem 2
n + -1.25 = -0.353553391 Simplifying n + -1.25 = -0.353553391 Reorder the terms: -1.25 + n = -0.353553391 Solving -1.25 + n = -0.353553391 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + n = -0.353553391 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + n = -0.353553391 + 1.25 n = -0.353553391 + 1.25 Combine like terms: -0.353553391 + 1.25 = 0.896446609 n = 0.896446609 Simplifying n = 0.896446609Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.603553391, 0.896446609}
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