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Simplifying n2 + 12n + 36 + -8 = 0 Reorder the terms: 36 + -8 + 12n + n2 = 0 Combine like terms: 36 + -8 = 28 28 + 12n + n2 = 0 Solving 28 + 12n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '-28' to each side of the equation. 28 + 12n + -28 + n2 = 0 + -28 Reorder the terms: 28 + -28 + 12n + n2 = 0 + -28 Combine like terms: 28 + -28 = 0 0 + 12n + n2 = 0 + -28 12n + n2 = 0 + -28 Combine like terms: 0 + -28 = -28 12n + n2 = -28 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 = -28 + 36 Reorder the terms: 36 + 12n + n2 = -28 + 36 Combine like terms: -28 + 36 = 8 36 + 12n + n2 = 8 Factor a perfect square on the left side: (n + 6)(n + 6) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (n + 6) equal to 2.828427125 and -2.828427125.Subproblem 1
n + 6 = 2.828427125 Simplifying n + 6 = 2.828427125 Reorder the terms: 6 + n = 2.828427125 Solving 6 + n = 2.828427125 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 = 2.828427125 + -6 Combine like terms: 6 + -6 = 0 0 + n = 2.828427125 + -6 n = 2.828427125 + -6 Combine like terms: 2.828427125 + -6 = -3.171572875 n = -3.171572875 Simplifying n = -3.171572875Subproblem 2
n + 6 = -2.828427125 Simplifying n + 6 = -2.828427125 Reorder the terms: 6 + n = -2.828427125 Solving 6 + n = -2.828427125 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 = -2.828427125 + -6 Combine like terms: 6 + -6 = 0 0 + n = -2.828427125 + -6 n = -2.828427125 + -6 Combine like terms: -2.828427125 + -6 = -8.828427125 n = -8.828427125 Simplifying n = -8.828427125Solution
The solution to the problem is based on the solutions from the subproblems. n = {-3.171572875, -8.828427125}
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