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Simplifying 6n2 + -2n + -3 + 5n2 + -9n + -6 = 0 Reorder the terms: -3 + -6 + -2n + -9n + 6n2 + 5n2 = 0 Combine like terms: -3 + -6 = -9 -9 + -2n + -9n + 6n2 + 5n2 = 0 Combine like terms: -2n + -9n = -11n -9 + -11n + 6n2 + 5n2 = 0 Combine like terms: 6n2 + 5n2 = 11n2 -9 + -11n + 11n2 = 0 Solving -9 + -11n + 11n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.8181818182 + -1n + n2 = 0 Move the constant term to the right: Add '0.8181818182' to each side of the equation. -0.8181818182 + -1n + 0.8181818182 + n2 = 0 + 0.8181818182 Reorder the terms: -0.8181818182 + 0.8181818182 + -1n + n2 = 0 + 0.8181818182 Combine like terms: -0.8181818182 + 0.8181818182 = 0.0000000000 0.0000000000 + -1n + n2 = 0 + 0.8181818182 -1n + n2 = 0 + 0.8181818182 Combine like terms: 0 + 0.8181818182 = 0.8181818182 -1n + n2 = 0.8181818182 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 0.8181818182 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 0.8181818182 + 0.25 Combine like terms: 0.8181818182 + 0.25 = 1.0681818182 0.25 + -1n + n2 = 1.0681818182 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 1.0681818182 Calculate the square root of the right side: 1.033528818 Break this problem into two subproblems by setting (n + -0.5) equal to 1.033528818 and -1.033528818.Subproblem 1
n + -0.5 = 1.033528818 Simplifying n + -0.5 = 1.033528818 Reorder the terms: -0.5 + n = 1.033528818 Solving -0.5 + n = 1.033528818 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 1.033528818 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.033528818 + 0.5 n = 1.033528818 + 0.5 Combine like terms: 1.033528818 + 0.5 = 1.533528818 n = 1.533528818 Simplifying n = 1.533528818Subproblem 2
n + -0.5 = -1.033528818 Simplifying n + -0.5 = -1.033528818 Reorder the terms: -0.5 + n = -1.033528818 Solving -0.5 + n = -1.033528818 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -1.033528818 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.033528818 + 0.5 n = -1.033528818 + 0.5 Combine like terms: -1.033528818 + 0.5 = -0.533528818 n = -0.533528818 Simplifying n = -0.533528818Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.533528818, -0.533528818}
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