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Simplifying 7n + 58 + -1n2 = 4 Reorder the terms: 58 + 7n + -1n2 = 4 Solving 58 + 7n + -1n2 = 4 Solving for variable 'n'. Reorder the terms: 58 + -4 + 7n + -1n2 = 4 + -4 Combine like terms: 58 + -4 = 54 54 + 7n + -1n2 = 4 + -4 Combine like terms: 4 + -4 = 0 54 + 7n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -54 + -7n + n2 = 0 Move the constant term to the right: Add '54' to each side of the equation. -54 + -7n + 54 + n2 = 0 + 54 Reorder the terms: -54 + 54 + -7n + n2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + -7n + n2 = 0 + 54 -7n + n2 = 0 + 54 Combine like terms: 0 + 54 = 54 -7n + n2 = 54 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 = 54 + 12.25 Reorder the terms: 12.25 + -7n + n2 = 54 + 12.25 Combine like terms: 54 + 12.25 = 66.25 12.25 + -7n + n2 = 66.25 Factor a perfect square on the left side: (n + -3.5)(n + -3.5) = 66.25 Calculate the square root of the right side: 8.139410298 Break this problem into two subproblems by setting (n + -3.5) equal to 8.139410298 and -8.139410298.Subproblem 1
n + -3.5 = 8.139410298 Simplifying n + -3.5 = 8.139410298 Reorder the terms: -3.5 + n = 8.139410298 Solving -3.5 + n = 8.139410298 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 = 8.139410298 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = 8.139410298 + 3.5 n = 8.139410298 + 3.5 Combine like terms: 8.139410298 + 3.5 = 11.639410298 n = 11.639410298 Simplifying n = 11.639410298Subproblem 2
n + -3.5 = -8.139410298 Simplifying n + -3.5 = -8.139410298 Reorder the terms: -3.5 + n = -8.139410298 Solving -3.5 + n = -8.139410298 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 = -8.139410298 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = -8.139410298 + 3.5 n = -8.139410298 + 3.5 Combine like terms: -8.139410298 + 3.5 = -4.639410298 n = -4.639410298 Simplifying n = -4.639410298Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.639410298, -4.639410298}
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