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Simplifying n2 + -129 = 7n Reorder the terms: -129 + n2 = 7n Solving -129 + n2 = 7n Solving for variable 'n'. Reorder the terms: -129 + -7n + n2 = 7n + -7n Combine like terms: 7n + -7n = 0 -129 + -7n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '129' to each side of the equation. -129 + -7n + 129 + n2 = 0 + 129 Reorder the terms: -129 + 129 + -7n + n2 = 0 + 129 Combine like terms: -129 + 129 = 0 0 + -7n + n2 = 0 + 129 -7n + n2 = 0 + 129 Combine like terms: 0 + 129 = 129 -7n + n2 = 129 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 = 129 + 12.25 Reorder the terms: 12.25 + -7n + n2 = 129 + 12.25 Combine like terms: 129 + 12.25 = 141.25 12.25 + -7n + n2 = 141.25 Factor a perfect square on the left side: (n + -3.5)(n + -3.5) = 141.25 Calculate the square root of the right side: 11.884864324 Break this problem into two subproblems by setting (n + -3.5) equal to 11.884864324 and -11.884864324.Subproblem 1
n + -3.5 = 11.884864324 Simplifying n + -3.5 = 11.884864324 Reorder the terms: -3.5 + n = 11.884864324 Solving -3.5 + n = 11.884864324 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 = 11.884864324 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = 11.884864324 + 3.5 n = 11.884864324 + 3.5 Combine like terms: 11.884864324 + 3.5 = 15.384864324 n = 15.384864324 Simplifying n = 15.384864324Subproblem 2
n + -3.5 = -11.884864324 Simplifying n + -3.5 = -11.884864324 Reorder the terms: -3.5 + n = -11.884864324 Solving -3.5 + n = -11.884864324 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 = -11.884864324 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = -11.884864324 + 3.5 n = -11.884864324 + 3.5 Combine like terms: -11.884864324 + 3.5 = -8.384864324 n = -8.384864324 Simplifying n = -8.384864324Solution
The solution to the problem is based on the solutions from the subproblems. n = {15.384864324, -8.384864324}
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