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Simplifying 3n2 + -2n + -6 = 9 Reorder the terms: -6 + -2n + 3n2 = 9 Solving -6 + -2n + 3n2 = 9 Solving for variable 'n'. Reorder the terms: -6 + -9 + -2n + 3n2 = 9 + -9 Combine like terms: -6 + -9 = -15 -15 + -2n + 3n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -15 + -2n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -5 + -0.6666666667n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.6666666667n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -0.6666666667n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.6666666667n + n2 = 0 + 5 -0.6666666667n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.6666666667n + n2 = 5 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 = 5 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667n + n2 = 5 + 0.1111111112 Combine like terms: 5 + 0.1111111112 = 5.1111111112 0.1111111112 + -0.6666666667n + n2 = 5.1111111112 Factor a perfect square on the left side: (n + -0.3333333334)(n + -0.3333333334) = 5.1111111112 Calculate the square root of the right side: 2.260776661 Break this problem into two subproblems by setting (n + -0.3333333334) equal to 2.260776661 and -2.260776661.Subproblem 1
n + -0.3333333334 = 2.260776661 Simplifying n + -0.3333333334 = 2.260776661 Reorder the terms: -0.3333333334 + n = 2.260776661 Solving -0.3333333334 + n = 2.260776661 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 = 2.260776661 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 2.260776661 + 0.3333333334 n = 2.260776661 + 0.3333333334 Combine like terms: 2.260776661 + 0.3333333334 = 2.5941099944 n = 2.5941099944 Simplifying n = 2.5941099944Subproblem 2
n + -0.3333333334 = -2.260776661 Simplifying n + -0.3333333334 = -2.260776661 Reorder the terms: -0.3333333334 + n = -2.260776661 Solving -0.3333333334 + n = -2.260776661 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 = -2.260776661 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -2.260776661 + 0.3333333334 n = -2.260776661 + 0.3333333334 Combine like terms: -2.260776661 + 0.3333333334 = -1.9274433276 n = -1.9274433276 Simplifying n = -1.9274433276Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.5941099944, -1.9274433276}
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