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Simplifying 4n2 + -16n + 4 = 0 Reorder the terms: 4 + -16n + 4n2 = 0 Solving 4 + -16n + 4n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4'. 4(1 + -4n + n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1 + -4n + n2)' equal to zero and attempt to solve: Simplifying 1 + -4n + n2 = 0 Solving 1 + -4n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -4n + -1 + n2 = 0 + -1 Reorder the terms: 1 + -1 + -4n + n2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -4n + n2 = 0 + -1 -4n + n2 = 0 + -1 Combine like terms: 0 + -1 = -1 -4n + n2 = -1 The n term is -4n. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4n + 4 + n2 = -1 + 4 Reorder the terms: 4 + -4n + n2 = -1 + 4 Combine like terms: -1 + 4 = 3 4 + -4n + n2 = 3 Factor a perfect square on the left side: (n + -2)(n + -2) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (n + -2) equal to 1.732050808 and -1.732050808.Subproblem 1
n + -2 = 1.732050808 Simplifying n + -2 = 1.732050808 Reorder the terms: -2 + n = 1.732050808 Solving -2 + n = 1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + n = 1.732050808 + 2 n = 1.732050808 + 2 Combine like terms: 1.732050808 + 2 = 3.732050808 n = 3.732050808 Simplifying n = 3.732050808Subproblem 2
n + -2 = -1.732050808 Simplifying n + -2 = -1.732050808 Reorder the terms: -2 + n = -1.732050808 Solving -2 + n = -1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + n = -1.732050808 + 2 n = -1.732050808 + 2 Combine like terms: -1.732050808 + 2 = 0.267949192 n = 0.267949192 Simplifying n = 0.267949192Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.732050808, 0.267949192}Solution
n = {3.732050808, 0.267949192}
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