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Simplifying n2 = 32 + -16n Solving n2 = 32 + -16n Solving for variable 'n'. Reorder the terms: -32 + 16n + n2 = 32 + -16n + -32 + 16n Reorder the terms: -32 + 16n + n2 = 32 + -32 + -16n + 16n Combine like terms: 32 + -32 = 0 -32 + 16n + n2 = 0 + -16n + 16n -32 + 16n + n2 = -16n + 16n Combine like terms: -16n + 16n = 0 -32 + 16n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + 16n + 32 + n2 = 0 + 32 Reorder the terms: -32 + 32 + 16n + n2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + 16n + n2 = 0 + 32 16n + n2 = 0 + 32 Combine like terms: 0 + 32 = 32 16n + n2 = 32 The n term is 16n. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16n + 64 + n2 = 32 + 64 Reorder the terms: 64 + 16n + n2 = 32 + 64 Combine like terms: 32 + 64 = 96 64 + 16n + n2 = 96 Factor a perfect square on the left side: (n + 8)(n + 8) = 96 Calculate the square root of the right side: 9.797958971 Break this problem into two subproblems by setting (n + 8) equal to 9.797958971 and -9.797958971.Subproblem 1
n + 8 = 9.797958971 Simplifying n + 8 = 9.797958971 Reorder the terms: 8 + n = 9.797958971 Solving 8 + n = 9.797958971 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = 9.797958971 + -8 Combine like terms: 8 + -8 = 0 0 + n = 9.797958971 + -8 n = 9.797958971 + -8 Combine like terms: 9.797958971 + -8 = 1.797958971 n = 1.797958971 Simplifying n = 1.797958971Subproblem 2
n + 8 = -9.797958971 Simplifying n + 8 = -9.797958971 Reorder the terms: 8 + n = -9.797958971 Solving 8 + n = -9.797958971 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = -9.797958971 + -8 Combine like terms: 8 + -8 = 0 0 + n = -9.797958971 + -8 n = -9.797958971 + -8 Combine like terms: -9.797958971 + -8 = -17.797958971 n = -17.797958971 Simplifying n = -17.797958971Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.797958971, -17.797958971}
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