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Simplifying 7n2 + -4n + -17 = 0 Reorder the terms: -17 + -4n + 7n2 = 0 Solving -17 + -4n + 7n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.428571429 + -0.5714285714n + n2 = 0 Move the constant term to the right: Add '2.428571429' to each side of the equation. -2.428571429 + -0.5714285714n + 2.428571429 + n2 = 0 + 2.428571429 Reorder the terms: -2.428571429 + 2.428571429 + -0.5714285714n + n2 = 0 + 2.428571429 Combine like terms: -2.428571429 + 2.428571429 = 0.000000000 0.000000000 + -0.5714285714n + n2 = 0 + 2.428571429 -0.5714285714n + n2 = 0 + 2.428571429 Combine like terms: 0 + 2.428571429 = 2.428571429 -0.5714285714n + n2 = 2.428571429 The n term is -0.5714285714n. Take half its coefficient (-0.2857142857). Square it (0.08163265305) and add it to both sides. Add '0.08163265305' to each side of the equation. -0.5714285714n + 0.08163265305 + n2 = 2.428571429 + 0.08163265305 Reorder the terms: 0.08163265305 + -0.5714285714n + n2 = 2.428571429 + 0.08163265305 Combine like terms: 2.428571429 + 0.08163265305 = 2.51020408205 0.08163265305 + -0.5714285714n + n2 = 2.51020408205 Factor a perfect square on the left side: (n + -0.2857142857)(n + -0.2857142857) = 2.51020408205 Calculate the square root of the right side: 1.584362358 Break this problem into two subproblems by setting (n + -0.2857142857) equal to 1.584362358 and -1.584362358.Subproblem 1
n + -0.2857142857 = 1.584362358 Simplifying n + -0.2857142857 = 1.584362358 Reorder the terms: -0.2857142857 + n = 1.584362358 Solving -0.2857142857 + n = 1.584362358 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + n = 1.584362358 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + n = 1.584362358 + 0.2857142857 n = 1.584362358 + 0.2857142857 Combine like terms: 1.584362358 + 0.2857142857 = 1.8700766437 n = 1.8700766437 Simplifying n = 1.8700766437Subproblem 2
n + -0.2857142857 = -1.584362358 Simplifying n + -0.2857142857 = -1.584362358 Reorder the terms: -0.2857142857 + n = -1.584362358 Solving -0.2857142857 + n = -1.584362358 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + n = -1.584362358 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + n = -1.584362358 + 0.2857142857 n = -1.584362358 + 0.2857142857 Combine like terms: -1.584362358 + 0.2857142857 = -1.2986480723 n = -1.2986480723 Simplifying n = -1.2986480723Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.8700766437, -1.2986480723}
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