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Simplifying 6n2 + -4n + -6 = 0 Reorder the terms: -6 + -4n + 6n2 = 0 Solving -6 + -4n + 6n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -2n + 3n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -2n + 3n2)' equal to zero and attempt to solve: Simplifying -3 + -2n + 3n2 = 0 Solving -3 + -2n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + -0.6666666667n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.6666666667n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -0.6666666667n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.6666666667n + n2 = 0 + 1 -0.6666666667n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.6666666667n + n2 = 1 The n term is -0.6666666667n. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667n + 0.1111111112 + n2 = 1 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667n + n2 = 1 + 0.1111111112 Combine like terms: 1 + 0.1111111112 = 1.1111111112 0.1111111112 + -0.6666666667n + n2 = 1.1111111112 Factor a perfect square on the left side: (n + -0.3333333334)(n + -0.3333333334) = 1.1111111112 Calculate the square root of the right side: 1.054092553 Break this problem into two subproblems by setting (n + -0.3333333334) equal to 1.054092553 and -1.054092553.Subproblem 1
n + -0.3333333334 = 1.054092553 Simplifying n + -0.3333333334 = 1.054092553 Reorder the terms: -0.3333333334 + n = 1.054092553 Solving -0.3333333334 + n = 1.054092553 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = 1.054092553 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 1.054092553 + 0.3333333334 n = 1.054092553 + 0.3333333334 Combine like terms: 1.054092553 + 0.3333333334 = 1.3874258864 n = 1.3874258864 Simplifying n = 1.3874258864Subproblem 2
n + -0.3333333334 = -1.054092553 Simplifying n + -0.3333333334 = -1.054092553 Reorder the terms: -0.3333333334 + n = -1.054092553 Solving -0.3333333334 + n = -1.054092553 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = -1.054092553 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -1.054092553 + 0.3333333334 n = -1.054092553 + 0.3333333334 Combine like terms: -1.054092553 + 0.3333333334 = -0.7207592196 n = -0.7207592196 Simplifying n = -0.7207592196Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.3874258864, -0.7207592196}Solution
n = {1.3874258864, -0.7207592196}
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