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Simplifying 9n2 + 8n + -19 = 0 Reorder the terms: -19 + 8n + 9n2 = 0 Solving -19 + 8n + 9n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.111111111 + 0.8888888889n + n2 = 0 Move the constant term to the right: Add '2.111111111' to each side of the equation. -2.111111111 + 0.8888888889n + 2.111111111 + n2 = 0 + 2.111111111 Reorder the terms: -2.111111111 + 2.111111111 + 0.8888888889n + n2 = 0 + 2.111111111 Combine like terms: -2.111111111 + 2.111111111 = 0.000000000 0.000000000 + 0.8888888889n + n2 = 0 + 2.111111111 0.8888888889n + n2 = 0 + 2.111111111 Combine like terms: 0 + 2.111111111 = 2.111111111 0.8888888889n + n2 = 2.111111111 The n term is 0.8888888889n. Take half its coefficient (0.4444444445). Square it (0.1975308642) and add it to both sides. Add '0.1975308642' to each side of the equation. 0.8888888889n + 0.1975308642 + n2 = 2.111111111 + 0.1975308642 Reorder the terms: 0.1975308642 + 0.8888888889n + n2 = 2.111111111 + 0.1975308642 Combine like terms: 2.111111111 + 0.1975308642 = 2.3086419752 0.1975308642 + 0.8888888889n + n2 = 2.3086419752 Factor a perfect square on the left side: (n + 0.4444444445)(n + 0.4444444445) = 2.3086419752 Calculate the square root of the right side: 1.519421592 Break this problem into two subproblems by setting (n + 0.4444444445) equal to 1.519421592 and -1.519421592.Subproblem 1
n + 0.4444444445 = 1.519421592 Simplifying n + 0.4444444445 = 1.519421592 Reorder the terms: 0.4444444445 + n = 1.519421592 Solving 0.4444444445 + n = 1.519421592 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + n = 1.519421592 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + n = 1.519421592 + -0.4444444445 n = 1.519421592 + -0.4444444445 Combine like terms: 1.519421592 + -0.4444444445 = 1.0749771475 n = 1.0749771475 Simplifying n = 1.0749771475Subproblem 2
n + 0.4444444445 = -1.519421592 Simplifying n + 0.4444444445 = -1.519421592 Reorder the terms: 0.4444444445 + n = -1.519421592 Solving 0.4444444445 + n = -1.519421592 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + n = -1.519421592 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + n = -1.519421592 + -0.4444444445 n = -1.519421592 + -0.4444444445 Combine like terms: -1.519421592 + -0.4444444445 = -1.9638660365 n = -1.9638660365 Simplifying n = -1.9638660365Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.0749771475, -1.9638660365}
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