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Simplifying n2 + -8n + 16 = 3 Reorder the terms: 16 + -8n + n2 = 3 Solving 16 + -8n + n2 = 3 Solving for variable 'n'. Reorder the terms: 16 + -3 + -8n + n2 = 3 + -3 Combine like terms: 16 + -3 = 13 13 + -8n + n2 = 3 + -3 Combine like terms: 3 + -3 = 0 13 + -8n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-13' to each side of the equation. 13 + -8n + -13 + n2 = 0 + -13 Reorder the terms: 13 + -13 + -8n + n2 = 0 + -13 Combine like terms: 13 + -13 = 0 0 + -8n + n2 = 0 + -13 -8n + n2 = 0 + -13 Combine like terms: 0 + -13 = -13 -8n + n2 = -13 The n term is -8n. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8n + 16 + n2 = -13 + 16 Reorder the terms: 16 + -8n + n2 = -13 + 16 Combine like terms: -13 + 16 = 3 16 + -8n + n2 = 3 Factor a perfect square on the left side: (n + -4)(n + -4) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (n + -4) equal to 1.732050808 and -1.732050808.Subproblem 1
n + -4 = 1.732050808 Simplifying n + -4 = 1.732050808 Reorder the terms: -4 + n = 1.732050808 Solving -4 + n = 1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = 1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + n = 1.732050808 + 4 n = 1.732050808 + 4 Combine like terms: 1.732050808 + 4 = 5.732050808 n = 5.732050808 Simplifying n = 5.732050808Subproblem 2
n + -4 = -1.732050808 Simplifying n + -4 = -1.732050808 Reorder the terms: -4 + n = -1.732050808 Solving -4 + n = -1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = -1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + n = -1.732050808 + 4 n = -1.732050808 + 4 Combine like terms: -1.732050808 + 4 = 2.267949192 n = 2.267949192 Simplifying n = 2.267949192Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.732050808, 2.267949192}
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