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Simplifying 7n2 + -32n + 20 = 0 Reorder the terms: 20 + -32n + 7n2 = 0 Solving 20 + -32n + 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'. 2.857142857 + -4.571428571n + n2 = 0 Move the constant term to the right: Add '-2.857142857' to each side of the equation. 2.857142857 + -4.571428571n + -2.857142857 + n2 = 0 + -2.857142857 Reorder the terms: 2.857142857 + -2.857142857 + -4.571428571n + n2 = 0 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + -4.571428571n + n2 = 0 + -2.857142857 -4.571428571n + n2 = 0 + -2.857142857 Combine like terms: 0 + -2.857142857 = -2.857142857 -4.571428571n + n2 = -2.857142857 The n term is -4.571428571n. Take half its coefficient (-2.285714286). Square it (5.224489797) and add it to both sides. Add '5.224489797' to each side of the equation. -4.571428571n + 5.224489797 + n2 = -2.857142857 + 5.224489797 Reorder the terms: 5.224489797 + -4.571428571n + n2 = -2.857142857 + 5.224489797 Combine like terms: -2.857142857 + 5.224489797 = 2.36734694 5.224489797 + -4.571428571n + n2 = 2.36734694 Factor a perfect square on the left side: (n + -2.285714286)(n + -2.285714286) = 2.36734694 Calculate the square root of the right side: 1.538618517 Break this problem into two subproblems by setting (n + -2.285714286) equal to 1.538618517 and -1.538618517.Subproblem 1
n + -2.285714286 = 1.538618517 Simplifying n + -2.285714286 = 1.538618517 Reorder the terms: -2.285714286 + n = 1.538618517 Solving -2.285714286 + n = 1.538618517 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.285714286' to each side of the equation. -2.285714286 + 2.285714286 + n = 1.538618517 + 2.285714286 Combine like terms: -2.285714286 + 2.285714286 = 0.000000000 0.000000000 + n = 1.538618517 + 2.285714286 n = 1.538618517 + 2.285714286 Combine like terms: 1.538618517 + 2.285714286 = 3.824332803 n = 3.824332803 Simplifying n = 3.824332803Subproblem 2
n + -2.285714286 = -1.538618517 Simplifying n + -2.285714286 = -1.538618517 Reorder the terms: -2.285714286 + n = -1.538618517 Solving -2.285714286 + n = -1.538618517 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.285714286' to each side of the equation. -2.285714286 + 2.285714286 + n = -1.538618517 + 2.285714286 Combine like terms: -2.285714286 + 2.285714286 = 0.000000000 0.000000000 + n = -1.538618517 + 2.285714286 n = -1.538618517 + 2.285714286 Combine like terms: -1.538618517 + 2.285714286 = 0.747095769 n = 0.747095769 Simplifying n = 0.747095769Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.824332803, 0.747095769}
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