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Simplifying (n + -3)(n + -3) = 3 Reorder the terms: (-3 + n)(n + -3) = 3 Reorder the terms: (-3 + n)(-3 + n) = 3 Multiply (-3 + n) * (-3 + n) (-3(-3 + n) + n(-3 + n)) = 3 ((-3 * -3 + n * -3) + n(-3 + n)) = 3 ((9 + -3n) + n(-3 + n)) = 3 (9 + -3n + (-3 * n + n * n)) = 3 (9 + -3n + (-3n + n2)) = 3 Combine like terms: -3n + -3n = -6n (9 + -6n + n2) = 3 Solving 9 + -6n + n2 = 3 Solving for variable 'n'. Reorder the terms: 9 + -3 + -6n + n2 = 3 + -3 Combine like terms: 9 + -3 = 6 6 + -6n + n2 = 3 + -3 Combine like terms: 3 + -3 = 0 6 + -6n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-6' to each side of the equation. 6 + -6n + -6 + n2 = 0 + -6 Reorder the terms: 6 + -6 + -6n + n2 = 0 + -6 Combine like terms: 6 + -6 = 0 0 + -6n + n2 = 0 + -6 -6n + n2 = 0 + -6 Combine like terms: 0 + -6 = -6 -6n + n2 = -6 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = -6 + 9 Reorder the terms: 9 + -6n + n2 = -6 + 9 Combine like terms: -6 + 9 = 3 9 + -6n + n2 = 3 Factor a perfect square on the left side: (n + -3)(n + -3) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (n + -3) equal to 1.732050808 and -1.732050808.Subproblem 1
n + -3 = 1.732050808 Simplifying n + -3 = 1.732050808 Reorder the terms: -3 + n = 1.732050808 Solving -3 + n = 1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 1.732050808 + 3 Combine like terms: -3 + 3 = 0 0 + n = 1.732050808 + 3 n = 1.732050808 + 3 Combine like terms: 1.732050808 + 3 = 4.732050808 n = 4.732050808 Simplifying n = 4.732050808Subproblem 2
n + -3 = -1.732050808 Simplifying n + -3 = -1.732050808 Reorder the terms: -3 + n = -1.732050808 Solving -3 + n = -1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -1.732050808 + 3 Combine like terms: -3 + 3 = 0 0 + n = -1.732050808 + 3 n = -1.732050808 + 3 Combine like terms: -1.732050808 + 3 = 1.267949192 n = 1.267949192 Simplifying n = 1.267949192Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.732050808, 1.267949192}
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