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Simplifying 3n2 + -2n + -14 = 4 Reorder the terms: -14 + -2n + 3n2 = 4 Solving -14 + -2n + 3n2 = 4 Solving for variable 'n'. Reorder the terms: -14 + -4 + -2n + 3n2 = 4 + -4 Combine like terms: -14 + -4 = -18 -18 + -2n + 3n2 = 4 + -4 Combine like terms: 4 + -4 = 0 -18 + -2n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6 + -0.6666666667n + n2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -0.6666666667n + 6 + n2 = 0 + 6 Reorder the terms: -6 + 6 + -0.6666666667n + n2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -0.6666666667n + n2 = 0 + 6 -0.6666666667n + n2 = 0 + 6 Combine like terms: 0 + 6 = 6 -0.6666666667n + n2 = 6 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 = 6 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667n + n2 = 6 + 0.1111111112 Combine like terms: 6 + 0.1111111112 = 6.1111111112 0.1111111112 + -0.6666666667n + n2 = 6.1111111112 Factor a perfect square on the left side: (n + -0.3333333334)(n + -0.3333333334) = 6.1111111112 Calculate the square root of the right side: 2.472066162 Break this problem into two subproblems by setting (n + -0.3333333334) equal to 2.472066162 and -2.472066162.Subproblem 1
n + -0.3333333334 = 2.472066162 Simplifying n + -0.3333333334 = 2.472066162 Reorder the terms: -0.3333333334 + n = 2.472066162 Solving -0.3333333334 + n = 2.472066162 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 = 2.472066162 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 2.472066162 + 0.3333333334 n = 2.472066162 + 0.3333333334 Combine like terms: 2.472066162 + 0.3333333334 = 2.8053994954 n = 2.8053994954 Simplifying n = 2.8053994954Subproblem 2
n + -0.3333333334 = -2.472066162 Simplifying n + -0.3333333334 = -2.472066162 Reorder the terms: -0.3333333334 + n = -2.472066162 Solving -0.3333333334 + n = -2.472066162 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 = -2.472066162 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -2.472066162 + 0.3333333334 n = -2.472066162 + 0.3333333334 Combine like terms: -2.472066162 + 0.3333333334 = -2.1387328286 n = -2.1387328286 Simplifying n = -2.1387328286Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.8053994954, -2.1387328286}
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