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Simplifying 2(1.5n + 4) * 6n = -7 Reorder the terms: 2(4 + 1.5n) * 6n = -7 Reorder the terms for easier multiplication: 2 * 6n(4 + 1.5n) = -7 Multiply 2 * 6 12n(4 + 1.5n) = -7 (4 * 12n + 1.5n * 12n) = -7 (48n + 18n2) = -7 Solving 48n + 18n2 = -7 Solving for variable 'n'. Reorder the terms: 7 + 48n + 18n2 = -7 + 7 Combine like terms: -7 + 7 = 0 7 + 48n + 18n2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.3888888889 + 2.666666667n + n2 = 0 Move the constant term to the right: Add '-0.3888888889' to each side of the equation. 0.3888888889 + 2.666666667n + -0.3888888889 + n2 = 0 + -0.3888888889 Reorder the terms: 0.3888888889 + -0.3888888889 + 2.666666667n + n2 = 0 + -0.3888888889 Combine like terms: 0.3888888889 + -0.3888888889 = 0.0000000000 0.0000000000 + 2.666666667n + n2 = 0 + -0.3888888889 2.666666667n + n2 = 0 + -0.3888888889 Combine like terms: 0 + -0.3888888889 = -0.3888888889 2.666666667n + n2 = -0.3888888889 The n term is 2.666666667n. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667n + 1.777777780 + n2 = -0.3888888889 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667n + n2 = -0.3888888889 + 1.777777780 Combine like terms: -0.3888888889 + 1.777777780 = 1.3888888911 1.777777780 + 2.666666667n + n2 = 1.3888888911 Factor a perfect square on the left side: (n + 1.333333334)(n + 1.333333334) = 1.3888888911 Calculate the square root of the right side: 1.178511303 Break this problem into two subproblems by setting (n + 1.333333334) equal to 1.178511303 and -1.178511303.Subproblem 1
n + 1.333333334 = 1.178511303 Simplifying n + 1.333333334 = 1.178511303 Reorder the terms: 1.333333334 + n = 1.178511303 Solving 1.333333334 + n = 1.178511303 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = 1.178511303 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = 1.178511303 + -1.333333334 n = 1.178511303 + -1.333333334 Combine like terms: 1.178511303 + -1.333333334 = -0.154822031 n = -0.154822031 Simplifying n = -0.154822031Subproblem 2
n + 1.333333334 = -1.178511303 Simplifying n + 1.333333334 = -1.178511303 Reorder the terms: 1.333333334 + n = -1.178511303 Solving 1.333333334 + n = -1.178511303 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = -1.178511303 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = -1.178511303 + -1.333333334 n = -1.178511303 + -1.333333334 Combine like terms: -1.178511303 + -1.333333334 = -2.511844637 n = -2.511844637 Simplifying n = -2.511844637Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.154822031, -2.511844637}
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