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Simplifying 6n2 + -42n + 71 = 0 Reorder the terms: 71 + -42n + 6n2 = 0 Solving 71 + -42n + 6n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 11.83333333 + -7n + n2 = 0 Move the constant term to the right: Add '-11.83333333' to each side of the equation. 11.83333333 + -7n + -11.83333333 + n2 = 0 + -11.83333333 Reorder the terms: 11.83333333 + -11.83333333 + -7n + n2 = 0 + -11.83333333 Combine like terms: 11.83333333 + -11.83333333 = 0.00000000 0.00000000 + -7n + n2 = 0 + -11.83333333 -7n + n2 = 0 + -11.83333333 Combine like terms: 0 + -11.83333333 = -11.83333333 -7n + n2 = -11.83333333 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 = -11.83333333 + 12.25 Reorder the terms: 12.25 + -7n + n2 = -11.83333333 + 12.25 Combine like terms: -11.83333333 + 12.25 = 0.41666667 12.25 + -7n + n2 = 0.41666667 Factor a perfect square on the left side: (n + -3.5)(n + -3.5) = 0.41666667 Calculate the square root of the right side: 0.645497227 Break this problem into two subproblems by setting (n + -3.5) equal to 0.645497227 and -0.645497227.Subproblem 1
n + -3.5 = 0.645497227 Simplifying n + -3.5 = 0.645497227 Reorder the terms: -3.5 + n = 0.645497227 Solving -3.5 + n = 0.645497227 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 = 0.645497227 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = 0.645497227 + 3.5 n = 0.645497227 + 3.5 Combine like terms: 0.645497227 + 3.5 = 4.145497227 n = 4.145497227 Simplifying n = 4.145497227Subproblem 2
n + -3.5 = -0.645497227 Simplifying n + -3.5 = -0.645497227 Reorder the terms: -3.5 + n = -0.645497227 Solving -3.5 + n = -0.645497227 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 = -0.645497227 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = -0.645497227 + 3.5 n = -0.645497227 + 3.5 Combine like terms: -0.645497227 + 3.5 = 2.854502773 n = 2.854502773 Simplifying n = 2.854502773Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.145497227, 2.854502773}
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