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Simplifying 3n2 + -4n + 5 + -2(n2 + 3n + -1) = 0 Reorder the terms: 3n2 + -4n + 5 + -2(-1 + 3n + n2) = 0 3n2 + -4n + 5 + (-1 * -2 + 3n * -2 + n2 * -2) = 0 3n2 + -4n + 5 + (2 + -6n + -2n2) = 0 Reorder the terms: 5 + 2 + -4n + -6n + 3n2 + -2n2 = 0 Combine like terms: 5 + 2 = 7 7 + -4n + -6n + 3n2 + -2n2 = 0 Combine like terms: -4n + -6n = -10n 7 + -10n + 3n2 + -2n2 = 0 Combine like terms: 3n2 + -2n2 = 1n2 7 + -10n + 1n2 = 0 Solving 7 + -10n + 1n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '-7' to each side of the equation. 7 + -10n + -7 + n2 = 0 + -7 Reorder the terms: 7 + -7 + -10n + n2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -10n + n2 = 0 + -7 -10n + n2 = 0 + -7 Combine like terms: 0 + -7 = -7 -10n + n2 = -7 The n term is -10n. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10n + 25 + n2 = -7 + 25 Reorder the terms: 25 + -10n + n2 = -7 + 25 Combine like terms: -7 + 25 = 18 25 + -10n + n2 = 18 Factor a perfect square on the left side: (n + -5)(n + -5) = 18 Calculate the square root of the right side: 4.242640687 Break this problem into two subproblems by setting (n + -5) equal to 4.242640687 and -4.242640687.Subproblem 1
n + -5 = 4.242640687 Simplifying n + -5 = 4.242640687 Reorder the terms: -5 + n = 4.242640687 Solving -5 + n = 4.242640687 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = 4.242640687 + 5 Combine like terms: -5 + 5 = 0 0 + n = 4.242640687 + 5 n = 4.242640687 + 5 Combine like terms: 4.242640687 + 5 = 9.242640687 n = 9.242640687 Simplifying n = 9.242640687Subproblem 2
n + -5 = -4.242640687 Simplifying n + -5 = -4.242640687 Reorder the terms: -5 + n = -4.242640687 Solving -5 + n = -4.242640687 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = -4.242640687 + 5 Combine like terms: -5 + 5 = 0 0 + n = -4.242640687 + 5 n = -4.242640687 + 5 Combine like terms: -4.242640687 + 5 = 0.757359313 n = 0.757359313 Simplifying n = 0.757359313Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.242640687, 0.757359313}
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