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Simplifying 2n2 + -16n + 23 = 7 Reorder the terms: 23 + -16n + 2n2 = 7 Solving 23 + -16n + 2n2 = 7 Solving for variable 'n'. Reorder the terms: 23 + -7 + -16n + 2n2 = 7 + -7 Combine like terms: 23 + -7 = 16 16 + -16n + 2n2 = 7 + -7 Combine like terms: 7 + -7 = 0 16 + -16n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(8 + -8n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(8 + -8n + n2)' equal to zero and attempt to solve: Simplifying 8 + -8n + n2 = 0 Solving 8 + -8n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-8' to each side of the equation. 8 + -8n + -8 + n2 = 0 + -8 Reorder the terms: 8 + -8 + -8n + n2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + -8n + n2 = 0 + -8 -8n + n2 = 0 + -8 Combine like terms: 0 + -8 = -8 -8n + n2 = -8 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 = -8 + 16 Reorder the terms: 16 + -8n + n2 = -8 + 16 Combine like terms: -8 + 16 = 8 16 + -8n + n2 = 8 Factor a perfect square on the left side: (n + -4)(n + -4) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (n + -4) equal to 2.828427125 and -2.828427125.Subproblem 1
n + -4 = 2.828427125 Simplifying n + -4 = 2.828427125 Reorder the terms: -4 + n = 2.828427125 Solving -4 + n = 2.828427125 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 = 2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + n = 2.828427125 + 4 n = 2.828427125 + 4 Combine like terms: 2.828427125 + 4 = 6.828427125 n = 6.828427125 Simplifying n = 6.828427125Subproblem 2
n + -4 = -2.828427125 Simplifying n + -4 = -2.828427125 Reorder the terms: -4 + n = -2.828427125 Solving -4 + n = -2.828427125 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 = -2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + n = -2.828427125 + 4 n = -2.828427125 + 4 Combine like terms: -2.828427125 + 4 = 1.171572875 n = 1.171572875 Simplifying n = 1.171572875Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.828427125, 1.171572875}Solution
n = {6.828427125, 1.171572875}
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