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Simplifying 2n(7 + -1n) = 1 (7 * 2n + -1n * 2n) = 1 (14n + -2n2) = 1 Solving 14n + -2n2 = 1 Solving for variable 'n'. Reorder the terms: -1 + 14n + -2n2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 14n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 0.5 + -7n + n2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -7n + -0.5 + n2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -7n + n2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -7n + n2 = 0 + -0.5 -7n + n2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -7n + n2 = -0.5 The n term is -7n. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7n + 12.25 + n2 = -0.5 + 12.25 Reorder the terms: 12.25 + -7n + n2 = -0.5 + 12.25 Combine like terms: -0.5 + 12.25 = 11.75 12.25 + -7n + n2 = 11.75 Factor a perfect square on the left side: (n + -3.5)(n + -3.5) = 11.75 Calculate the square root of the right side: 3.4278273 Break this problem into two subproblems by setting (n + -3.5) equal to 3.4278273 and -3.4278273.Subproblem 1
n + -3.5 = 3.4278273 Simplifying n + -3.5 = 3.4278273 Reorder the terms: -3.5 + n = 3.4278273 Solving -3.5 + n = 3.4278273 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + n = 3.4278273 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = 3.4278273 + 3.5 n = 3.4278273 + 3.5 Combine like terms: 3.4278273 + 3.5 = 6.9278273 n = 6.9278273 Simplifying n = 6.9278273Subproblem 2
n + -3.5 = -3.4278273 Simplifying n + -3.5 = -3.4278273 Reorder the terms: -3.5 + n = -3.4278273 Solving -3.5 + n = -3.4278273 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + n = -3.4278273 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = -3.4278273 + 3.5 n = -3.4278273 + 3.5 Combine like terms: -3.4278273 + 3.5 = 0.0721727 n = 0.0721727 Simplifying n = 0.0721727Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.9278273, 0.0721727}
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