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Simplifying n2 + -22n + 102 = 0 Reorder the terms: 102 + -22n + n2 = 0 Solving 102 + -22n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '-102' to each side of the equation. 102 + -22n + -102 + n2 = 0 + -102 Reorder the terms: 102 + -102 + -22n + n2 = 0 + -102 Combine like terms: 102 + -102 = 0 0 + -22n + n2 = 0 + -102 -22n + n2 = 0 + -102 Combine like terms: 0 + -102 = -102 -22n + n2 = -102 The n term is -22n. Take half its coefficient (-11). Square it (121) and add it to both sides. Add '121' to each side of the equation. -22n + 121 + n2 = -102 + 121 Reorder the terms: 121 + -22n + n2 = -102 + 121 Combine like terms: -102 + 121 = 19 121 + -22n + n2 = 19 Factor a perfect square on the left side: (n + -11)(n + -11) = 19 Calculate the square root of the right side: 4.358898944 Break this problem into two subproblems by setting (n + -11) equal to 4.358898944 and -4.358898944.Subproblem 1
n + -11 = 4.358898944 Simplifying n + -11 = 4.358898944 Reorder the terms: -11 + n = 4.358898944 Solving -11 + n = 4.358898944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + n = 4.358898944 + 11 Combine like terms: -11 + 11 = 0 0 + n = 4.358898944 + 11 n = 4.358898944 + 11 Combine like terms: 4.358898944 + 11 = 15.358898944 n = 15.358898944 Simplifying n = 15.358898944Subproblem 2
n + -11 = -4.358898944 Simplifying n + -11 = -4.358898944 Reorder the terms: -11 + n = -4.358898944 Solving -11 + n = -4.358898944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + n = -4.358898944 + 11 Combine like terms: -11 + 11 = 0 0 + n = -4.358898944 + 11 n = -4.358898944 + 11 Combine like terms: -4.358898944 + 11 = 6.641101056 n = 6.641101056 Simplifying n = 6.641101056Solution
The solution to the problem is based on the solutions from the subproblems. n = {15.358898944, 6.641101056}
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