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Simplifying (-6 + -1n + 2n2) + (8 + -3n + -19n2) = 0 Remove parenthesis around (-6 + -1n + 2n2) -6 + -1n + 2n2 + (8 + -3n + -19n2) = 0 Remove parenthesis around (8 + -3n + -19n2) -6 + -1n + 2n2 + 8 + -3n + -19n2 = 0 Reorder the terms: -6 + 8 + -1n + -3n + 2n2 + -19n2 = 0 Combine like terms: -6 + 8 = 2 2 + -1n + -3n + 2n2 + -19n2 = 0 Combine like terms: -1n + -3n = -4n 2 + -4n + 2n2 + -19n2 = 0 Combine like terms: 2n2 + -19n2 = -17n2 2 + -4n + -17n2 = 0 Solving 2 + -4n + -17n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -17 the coefficient of the squared term: Divide each side by '-17'. -0.1176470588 + 0.2352941176n + n2 = 0 Move the constant term to the right: Add '0.1176470588' to each side of the equation. -0.1176470588 + 0.2352941176n + 0.1176470588 + n2 = 0 + 0.1176470588 Reorder the terms: -0.1176470588 + 0.1176470588 + 0.2352941176n + n2 = 0 + 0.1176470588 Combine like terms: -0.1176470588 + 0.1176470588 = 0.0000000000 0.0000000000 + 0.2352941176n + n2 = 0 + 0.1176470588 0.2352941176n + n2 = 0 + 0.1176470588 Combine like terms: 0 + 0.1176470588 = 0.1176470588 0.2352941176n + n2 = 0.1176470588 The n term is 0.2352941176n. Take half its coefficient (0.1176470588). Square it (0.01384083044) and add it to both sides. Add '0.01384083044' to each side of the equation. 0.2352941176n + 0.01384083044 + n2 = 0.1176470588 + 0.01384083044 Reorder the terms: 0.01384083044 + 0.2352941176n + n2 = 0.1176470588 + 0.01384083044 Combine like terms: 0.1176470588 + 0.01384083044 = 0.13148788924 0.01384083044 + 0.2352941176n + n2 = 0.13148788924 Factor a perfect square on the left side: (n + 0.1176470588)(n + 0.1176470588) = 0.13148788924 Calculate the square root of the right side: 0.362612588 Break this problem into two subproblems by setting (n + 0.1176470588) equal to 0.362612588 and -0.362612588.Subproblem 1
n + 0.1176470588 = 0.362612588 Simplifying n + 0.1176470588 = 0.362612588 Reorder the terms: 0.1176470588 + n = 0.362612588 Solving 0.1176470588 + n = 0.362612588 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1176470588' to each side of the equation. 0.1176470588 + -0.1176470588 + n = 0.362612588 + -0.1176470588 Combine like terms: 0.1176470588 + -0.1176470588 = 0.0000000000 0.0000000000 + n = 0.362612588 + -0.1176470588 n = 0.362612588 + -0.1176470588 Combine like terms: 0.362612588 + -0.1176470588 = 0.2449655292 n = 0.2449655292 Simplifying n = 0.2449655292Subproblem 2
n + 0.1176470588 = -0.362612588 Simplifying n + 0.1176470588 = -0.362612588 Reorder the terms: 0.1176470588 + n = -0.362612588 Solving 0.1176470588 + n = -0.362612588 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1176470588' to each side of the equation. 0.1176470588 + -0.1176470588 + n = -0.362612588 + -0.1176470588 Combine like terms: 0.1176470588 + -0.1176470588 = 0.0000000000 0.0000000000 + n = -0.362612588 + -0.1176470588 n = -0.362612588 + -0.1176470588 Combine like terms: -0.362612588 + -0.1176470588 = -0.4802596468 n = -0.4802596468 Simplifying n = -0.4802596468Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.2449655292, -0.4802596468}
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