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Simplifying 3n(n + -6) + 4 = 5n + -6 Reorder the terms: 3n(-6 + n) + 4 = 5n + -6 (-6 * 3n + n * 3n) + 4 = 5n + -6 (-18n + 3n2) + 4 = 5n + -6 Reorder the terms: 4 + -18n + 3n2 = 5n + -6 Reorder the terms: 4 + -18n + 3n2 = -6 + 5n Solving 4 + -18n + 3n2 = -6 + 5n Solving for variable 'n'. Reorder the terms: 4 + 6 + -18n + -5n + 3n2 = -6 + 5n + 6 + -5n Combine like terms: 4 + 6 = 10 10 + -18n + -5n + 3n2 = -6 + 5n + 6 + -5n Combine like terms: -18n + -5n = -23n 10 + -23n + 3n2 = -6 + 5n + 6 + -5n Reorder the terms: 10 + -23n + 3n2 = -6 + 6 + 5n + -5n Combine like terms: -6 + 6 = 0 10 + -23n + 3n2 = 0 + 5n + -5n 10 + -23n + 3n2 = 5n + -5n Combine like terms: 5n + -5n = 0 10 + -23n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 3.333333333 + -7.666666667n + n2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + -7.666666667n + -3.333333333 + n2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + -7.666666667n + n2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + -7.666666667n + n2 = 0 + -3.333333333 -7.666666667n + n2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 -7.666666667n + n2 = -3.333333333 The n term is -7.666666667n. Take half its coefficient (-3.833333334). Square it (14.69444445) and add it to both sides. Add '14.69444445' to each side of the equation. -7.666666667n + 14.69444445 + n2 = -3.333333333 + 14.69444445 Reorder the terms: 14.69444445 + -7.666666667n + n2 = -3.333333333 + 14.69444445 Combine like terms: -3.333333333 + 14.69444445 = 11.361111117 14.69444445 + -7.666666667n + n2 = 11.361111117 Factor a perfect square on the left side: (n + -3.833333334)(n + -3.833333334) = 11.361111117 Calculate the square root of the right side: 3.370624737 Break this problem into two subproblems by setting (n + -3.833333334) equal to 3.370624737 and -3.370624737.Subproblem 1
n + -3.833333334 = 3.370624737 Simplifying n + -3.833333334 = 3.370624737 Reorder the terms: -3.833333334 + n = 3.370624737 Solving -3.833333334 + n = 3.370624737 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.833333334' to each side of the equation. -3.833333334 + 3.833333334 + n = 3.370624737 + 3.833333334 Combine like terms: -3.833333334 + 3.833333334 = 0.000000000 0.000000000 + n = 3.370624737 + 3.833333334 n = 3.370624737 + 3.833333334 Combine like terms: 3.370624737 + 3.833333334 = 7.203958071 n = 7.203958071 Simplifying n = 7.203958071Subproblem 2
n + -3.833333334 = -3.370624737 Simplifying n + -3.833333334 = -3.370624737 Reorder the terms: -3.833333334 + n = -3.370624737 Solving -3.833333334 + n = -3.370624737 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.833333334' to each side of the equation. -3.833333334 + 3.833333334 + n = -3.370624737 + 3.833333334 Combine like terms: -3.833333334 + 3.833333334 = 0.000000000 0.000000000 + n = -3.370624737 + 3.833333334 n = -3.370624737 + 3.833333334 Combine like terms: -3.370624737 + 3.833333334 = 0.462708597 n = 0.462708597 Simplifying n = 0.462708597Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.203958071, 0.462708597}
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