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Simplifying 3n2 + 12n + 9 = -1 Reorder the terms: 9 + 12n + 3n2 = -1 Solving 9 + 12n + 3n2 = -1 Solving for variable 'n'. Reorder the terms: 9 + 1 + 12n + 3n2 = -1 + 1 Combine like terms: 9 + 1 = 10 10 + 12n + 3n2 = -1 + 1 Combine like terms: -1 + 1 = 0 10 + 12n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 3.333333333 + 4n + n2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + 4n + -3.333333333 + n2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + 4n + n2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + 4n + n2 = 0 + -3.333333333 4n + n2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 4n + n2 = -3.333333333 The n term is 4n. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4n + 4 + n2 = -3.333333333 + 4 Reorder the terms: 4 + 4n + n2 = -3.333333333 + 4 Combine like terms: -3.333333333 + 4 = 0.666666667 4 + 4n + n2 = 0.666666667 Factor a perfect square on the left side: (n + 2)(n + 2) = 0.666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (n + 2) equal to 0.816496581 and -0.816496581.Subproblem 1
n + 2 = 0.816496581 Simplifying n + 2 = 0.816496581 Reorder the terms: 2 + n = 0.816496581 Solving 2 + n = 0.816496581 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 0.816496581 + -2 Combine like terms: 2 + -2 = 0 0 + n = 0.816496581 + -2 n = 0.816496581 + -2 Combine like terms: 0.816496581 + -2 = -1.183503419 n = -1.183503419 Simplifying n = -1.183503419Subproblem 2
n + 2 = -0.816496581 Simplifying n + 2 = -0.816496581 Reorder the terms: 2 + n = -0.816496581 Solving 2 + n = -0.816496581 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -0.816496581 + -2 Combine like terms: 2 + -2 = 0 0 + n = -0.816496581 + -2 n = -0.816496581 + -2 Combine like terms: -0.816496581 + -2 = -2.816496581 n = -2.816496581 Simplifying n = -2.816496581Solution
The solution to the problem is based on the solutions from the subproblems. n = {-1.183503419, -2.816496581}
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