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Simplifying 91 = n2 + -1n Reorder the terms: 91 = -1n + n2 Solving 91 = -1n + n2 Solving for variable 'n'. Reorder the terms: 91 + n + -1n2 = -1n + n + n2 + -1n2 Combine like terms: -1n + n = 0 91 + n + -1n2 = 0 + n2 + -1n2 91 + n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 91 + n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -91 + -1n + n2 = 0 Move the constant term to the right: Add '91' to each side of the equation. -91 + -1n + 91 + n2 = 0 + 91 Reorder the terms: -91 + 91 + -1n + n2 = 0 + 91 Combine like terms: -91 + 91 = 0 0 + -1n + n2 = 0 + 91 -1n + n2 = 0 + 91 Combine like terms: 0 + 91 = 91 -1n + n2 = 91 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 91 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 91 + 0.25 Combine like terms: 91 + 0.25 = 91.25 0.25 + -1n + n2 = 91.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 91.25 Calculate the square root of the right side: 9.552486587 Break this problem into two subproblems by setting (n + 0.5) equal to 9.552486587 and -9.552486587.Subproblem 1
n + 0.5 = 9.552486587 Simplifying n + 0.5 = 9.552486587 Reorder the terms: 0.5 + n = 9.552486587 Solving 0.5 + n = 9.552486587 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 9.552486587 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 9.552486587 + -0.5 n = 9.552486587 + -0.5 Combine like terms: 9.552486587 + -0.5 = 9.052486587 n = 9.052486587 Simplifying n = 9.052486587Subproblem 2
n + 0.5 = -9.552486587 Simplifying n + 0.5 = -9.552486587 Reorder the terms: 0.5 + n = -9.552486587 Solving 0.5 + n = -9.552486587 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -9.552486587 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -9.552486587 + -0.5 n = -9.552486587 + -0.5 Combine like terms: -9.552486587 + -0.5 = -10.052486587 n = -10.052486587 Simplifying n = -10.052486587Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.052486587, -10.052486587}
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