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Simplifying 2n2 + -48n + -288 = 0 Reorder the terms: -288 + -48n + 2n2 = 0 Solving -288 + -48n + 2n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-144 + -24n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-144 + -24n + n2)' equal to zero and attempt to solve: Simplifying -144 + -24n + n2 = 0 Solving -144 + -24n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '144' to each side of the equation. -144 + -24n + 144 + n2 = 0 + 144 Reorder the terms: -144 + 144 + -24n + n2 = 0 + 144 Combine like terms: -144 + 144 = 0 0 + -24n + n2 = 0 + 144 -24n + n2 = 0 + 144 Combine like terms: 0 + 144 = 144 -24n + n2 = 144 The n term is -24n. Take half its coefficient (-12). Square it (144) and add it to both sides. Add '144' to each side of the equation. -24n + 144 + n2 = 144 + 144 Reorder the terms: 144 + -24n + n2 = 144 + 144 Combine like terms: 144 + 144 = 288 144 + -24n + n2 = 288 Factor a perfect square on the left side: (n + -12)(n + -12) = 288 Calculate the square root of the right side: 16.970562748 Break this problem into two subproblems by setting (n + -12) equal to 16.970562748 and -16.970562748.Subproblem 1
n + -12 = 16.970562748 Simplifying n + -12 = 16.970562748 Reorder the terms: -12 + n = 16.970562748 Solving -12 + n = 16.970562748 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + n = 16.970562748 + 12 Combine like terms: -12 + 12 = 0 0 + n = 16.970562748 + 12 n = 16.970562748 + 12 Combine like terms: 16.970562748 + 12 = 28.970562748 n = 28.970562748 Simplifying n = 28.970562748Subproblem 2
n + -12 = -16.970562748 Simplifying n + -12 = -16.970562748 Reorder the terms: -12 + n = -16.970562748 Solving -12 + n = -16.970562748 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + n = -16.970562748 + 12 Combine like terms: -12 + 12 = 0 0 + n = -16.970562748 + 12 n = -16.970562748 + 12 Combine like terms: -16.970562748 + 12 = -4.970562748 n = -4.970562748 Simplifying n = -4.970562748Solution
The solution to the problem is based on the solutions from the subproblems. n = {28.970562748, -4.970562748}Solution
n = {28.970562748, -4.970562748}
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