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Simplifying 7n2 + -17n + 3 = 0 Reorder the terms: 3 + -17n + 7n2 = 0 Solving 3 + -17n + 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'. 0.4285714286 + -2.428571429n + n2 = 0 Move the constant term to the right: Add '-0.4285714286' to each side of the equation. 0.4285714286 + -2.428571429n + -0.4285714286 + n2 = 0 + -0.4285714286 Reorder the terms: 0.4285714286 + -0.4285714286 + -2.428571429n + n2 = 0 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + -2.428571429n + n2 = 0 + -0.4285714286 -2.428571429n + n2 = 0 + -0.4285714286 Combine like terms: 0 + -0.4285714286 = -0.4285714286 -2.428571429n + n2 = -0.4285714286 The n term is -2.428571429n. Take half its coefficient (-1.214285715). Square it (1.474489798) and add it to both sides. Add '1.474489798' to each side of the equation. -2.428571429n + 1.474489798 + n2 = -0.4285714286 + 1.474489798 Reorder the terms: 1.474489798 + -2.428571429n + n2 = -0.4285714286 + 1.474489798 Combine like terms: -0.4285714286 + 1.474489798 = 1.0459183694 1.474489798 + -2.428571429n + n2 = 1.0459183694 Factor a perfect square on the left side: (n + -1.214285715)(n + -1.214285715) = 1.0459183694 Calculate the square root of the right side: 1.022701506 Break this problem into two subproblems by setting (n + -1.214285715) equal to 1.022701506 and -1.022701506.Subproblem 1
n + -1.214285715 = 1.022701506 Simplifying n + -1.214285715 = 1.022701506 Reorder the terms: -1.214285715 + n = 1.022701506 Solving -1.214285715 + n = 1.022701506 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.214285715' to each side of the equation. -1.214285715 + 1.214285715 + n = 1.022701506 + 1.214285715 Combine like terms: -1.214285715 + 1.214285715 = 0.000000000 0.000000000 + n = 1.022701506 + 1.214285715 n = 1.022701506 + 1.214285715 Combine like terms: 1.022701506 + 1.214285715 = 2.236987221 n = 2.236987221 Simplifying n = 2.236987221Subproblem 2
n + -1.214285715 = -1.022701506 Simplifying n + -1.214285715 = -1.022701506 Reorder the terms: -1.214285715 + n = -1.022701506 Solving -1.214285715 + n = -1.022701506 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.214285715' to each side of the equation. -1.214285715 + 1.214285715 + n = -1.022701506 + 1.214285715 Combine like terms: -1.214285715 + 1.214285715 = 0.000000000 0.000000000 + n = -1.022701506 + 1.214285715 n = -1.022701506 + 1.214285715 Combine like terms: -1.022701506 + 1.214285715 = 0.191584209 n = 0.191584209 Simplifying n = 0.191584209Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.236987221, 0.191584209}
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