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Simplifying n2 + -68 = 18n Reorder the terms: -68 + n2 = 18n Solving -68 + n2 = 18n Solving for variable 'n'. Reorder the terms: -68 + -18n + n2 = 18n + -18n Combine like terms: 18n + -18n = 0 -68 + -18n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '68' to each side of the equation. -68 + -18n + 68 + n2 = 0 + 68 Reorder the terms: -68 + 68 + -18n + n2 = 0 + 68 Combine like terms: -68 + 68 = 0 0 + -18n + n2 = 0 + 68 -18n + n2 = 0 + 68 Combine like terms: 0 + 68 = 68 -18n + n2 = 68 The n term is -18n. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18n + 81 + n2 = 68 + 81 Reorder the terms: 81 + -18n + n2 = 68 + 81 Combine like terms: 68 + 81 = 149 81 + -18n + n2 = 149 Factor a perfect square on the left side: (n + -9)(n + -9) = 149 Calculate the square root of the right side: 12.206555616 Break this problem into two subproblems by setting (n + -9) equal to 12.206555616 and -12.206555616.Subproblem 1
n + -9 = 12.206555616 Simplifying n + -9 = 12.206555616 Reorder the terms: -9 + n = 12.206555616 Solving -9 + n = 12.206555616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = 12.206555616 + 9 Combine like terms: -9 + 9 = 0 0 + n = 12.206555616 + 9 n = 12.206555616 + 9 Combine like terms: 12.206555616 + 9 = 21.206555616 n = 21.206555616 Simplifying n = 21.206555616Subproblem 2
n + -9 = -12.206555616 Simplifying n + -9 = -12.206555616 Reorder the terms: -9 + n = -12.206555616 Solving -9 + n = -12.206555616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = -12.206555616 + 9 Combine like terms: -9 + 9 = 0 0 + n = -12.206555616 + 9 n = -12.206555616 + 9 Combine like terms: -12.206555616 + 9 = -3.206555616 n = -3.206555616 Simplifying n = -3.206555616Solution
The solution to the problem is based on the solutions from the subproblems. n = {21.206555616, -3.206555616}
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