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Simplifying n2 + -12n + 23 = 0 Reorder the terms: 23 + -12n + n2 = 0 Solving 23 + -12n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '-23' to each side of the equation. 23 + -12n + -23 + n2 = 0 + -23 Reorder the terms: 23 + -23 + -12n + n2 = 0 + -23 Combine like terms: 23 + -23 = 0 0 + -12n + n2 = 0 + -23 -12n + n2 = 0 + -23 Combine like terms: 0 + -23 = -23 -12n + n2 = -23 The n term is -12n. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12n + 36 + n2 = -23 + 36 Reorder the terms: 36 + -12n + n2 = -23 + 36 Combine like terms: -23 + 36 = 13 36 + -12n + n2 = 13 Factor a perfect square on the left side: (n + -6)(n + -6) = 13 Calculate the square root of the right side: 3.605551275 Break this problem into two subproblems by setting (n + -6) equal to 3.605551275 and -3.605551275.Subproblem 1
n + -6 = 3.605551275 Simplifying n + -6 = 3.605551275 Reorder the terms: -6 + n = 3.605551275 Solving -6 + n = 3.605551275 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = 3.605551275 + 6 Combine like terms: -6 + 6 = 0 0 + n = 3.605551275 + 6 n = 3.605551275 + 6 Combine like terms: 3.605551275 + 6 = 9.605551275 n = 9.605551275 Simplifying n = 9.605551275Subproblem 2
n + -6 = -3.605551275 Simplifying n + -6 = -3.605551275 Reorder the terms: -6 + n = -3.605551275 Solving -6 + n = -3.605551275 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = -3.605551275 + 6 Combine like terms: -6 + 6 = 0 0 + n = -3.605551275 + 6 n = -3.605551275 + 6 Combine like terms: -3.605551275 + 6 = 2.394448725 n = 2.394448725 Simplifying n = 2.394448725Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.605551275, 2.394448725}
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