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Simplifying 9n2 + -6n + -12 = 2 Reorder the terms: -12 + -6n + 9n2 = 2 Solving -12 + -6n + 9n2 = 2 Solving for variable 'n'. Reorder the terms: -12 + -2 + -6n + 9n2 = 2 + -2 Combine like terms: -12 + -2 = -14 -14 + -6n + 9n2 = 2 + -2 Combine like terms: 2 + -2 = 0 -14 + -6n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.555555556 + -0.6666666667n + n2 = 0 Move the constant term to the right: Add '1.555555556' to each side of the equation. -1.555555556 + -0.6666666667n + 1.555555556 + n2 = 0 + 1.555555556 Reorder the terms: -1.555555556 + 1.555555556 + -0.6666666667n + n2 = 0 + 1.555555556 Combine like terms: -1.555555556 + 1.555555556 = 0.000000000 0.000000000 + -0.6666666667n + n2 = 0 + 1.555555556 -0.6666666667n + n2 = 0 + 1.555555556 Combine like terms: 0 + 1.555555556 = 1.555555556 -0.6666666667n + n2 = 1.555555556 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 = 1.555555556 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667n + n2 = 1.555555556 + 0.1111111112 Combine like terms: 1.555555556 + 0.1111111112 = 1.6666666672 0.1111111112 + -0.6666666667n + n2 = 1.6666666672 Factor a perfect square on the left side: (n + -0.3333333334)(n + -0.3333333334) = 1.6666666672 Calculate the square root of the right side: 1.290994449 Break this problem into two subproblems by setting (n + -0.3333333334) equal to 1.290994449 and -1.290994449.Subproblem 1
n + -0.3333333334 = 1.290994449 Simplifying n + -0.3333333334 = 1.290994449 Reorder the terms: -0.3333333334 + n = 1.290994449 Solving -0.3333333334 + n = 1.290994449 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 = 1.290994449 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 1.290994449 + 0.3333333334 n = 1.290994449 + 0.3333333334 Combine like terms: 1.290994449 + 0.3333333334 = 1.6243277824 n = 1.6243277824 Simplifying n = 1.6243277824Subproblem 2
n + -0.3333333334 = -1.290994449 Simplifying n + -0.3333333334 = -1.290994449 Reorder the terms: -0.3333333334 + n = -1.290994449 Solving -0.3333333334 + n = -1.290994449 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 = -1.290994449 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -1.290994449 + 0.3333333334 n = -1.290994449 + 0.3333333334 Combine like terms: -1.290994449 + 0.3333333334 = -0.9576611156 n = -0.9576611156 Simplifying n = -0.9576611156Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.6243277824, -0.9576611156}
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