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Simplifying 0 = 16n2 + 112n + -120 Reorder the terms: 0 = -120 + 112n + 16n2 Solving 0 = -120 + 112n + 16n2 Solving for variable 'n'. Combine like terms: 0 + 120 = 120 120 + -112n + -16n2 = -120 + 112n + 16n2 + 120 + -112n + -16n2 Reorder the terms: 120 + -112n + -16n2 = -120 + 120 + 112n + -112n + 16n2 + -16n2 Combine like terms: -120 + 120 = 0 120 + -112n + -16n2 = 0 + 112n + -112n + 16n2 + -16n2 120 + -112n + -16n2 = 112n + -112n + 16n2 + -16n2 Combine like terms: 112n + -112n = 0 120 + -112n + -16n2 = 0 + 16n2 + -16n2 120 + -112n + -16n2 = 16n2 + -16n2 Combine like terms: 16n2 + -16n2 = 0 120 + -112n + -16n2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(15 + -14n + -2n2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(15 + -14n + -2n2)' equal to zero and attempt to solve: Simplifying 15 + -14n + -2n2 = 0 Solving 15 + -14n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -7.5 + 7n + n2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 7n + 7.5 + n2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 7n + n2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 7n + n2 = 0 + 7.5 7n + n2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 7n + n2 = 7.5 The n term is 7n. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7n + 12.25 + n2 = 7.5 + 12.25 Reorder the terms: 12.25 + 7n + n2 = 7.5 + 12.25 Combine like terms: 7.5 + 12.25 = 19.75 12.25 + 7n + n2 = 19.75 Factor a perfect square on the left side: (n + 3.5)(n + 3.5) = 19.75 Calculate the square root of the right side: 4.444097209 Break this problem into two subproblems by setting (n + 3.5) equal to 4.444097209 and -4.444097209.Subproblem 1
n + 3.5 = 4.444097209 Simplifying n + 3.5 = 4.444097209 Reorder the terms: 3.5 + n = 4.444097209 Solving 3.5 + n = 4.444097209 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 4.444097209 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 4.444097209 + -3.5 n = 4.444097209 + -3.5 Combine like terms: 4.444097209 + -3.5 = 0.944097209 n = 0.944097209 Simplifying n = 0.944097209Subproblem 2
n + 3.5 = -4.444097209 Simplifying n + 3.5 = -4.444097209 Reorder the terms: 3.5 + n = -4.444097209 Solving 3.5 + n = -4.444097209 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -4.444097209 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -4.444097209 + -3.5 n = -4.444097209 + -3.5 Combine like terms: -4.444097209 + -3.5 = -7.944097209 n = -7.944097209 Simplifying n = -7.944097209Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.944097209, -7.944097209}Solution
n = {0.944097209, -7.944097209}
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