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Simplifying 6n2 + 29n + -5 = 1 Reorder the terms: -5 + 29n + 6n2 = 1 Solving -5 + 29n + 6n2 = 1 Solving for variable 'n'. Reorder the terms: -5 + -1 + 29n + 6n2 = 1 + -1 Combine like terms: -5 + -1 = -6 -6 + 29n + 6n2 = 1 + -1 Combine like terms: 1 + -1 = 0 -6 + 29n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1 + 4.833333333n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 4.833333333n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + 4.833333333n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 4.833333333n + n2 = 0 + 1 4.833333333n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 4.833333333n + n2 = 1 The n term is 4.833333333n. Take half its coefficient (2.416666667). Square it (5.840277779) and add it to both sides. Add '5.840277779' to each side of the equation. 4.833333333n + 5.840277779 + n2 = 1 + 5.840277779 Reorder the terms: 5.840277779 + 4.833333333n + n2 = 1 + 5.840277779 Combine like terms: 1 + 5.840277779 = 6.840277779 5.840277779 + 4.833333333n + n2 = 6.840277779 Factor a perfect square on the left side: (n + 2.416666667)(n + 2.416666667) = 6.840277779 Calculate the square root of the right side: 2.615392471 Break this problem into two subproblems by setting (n + 2.416666667) equal to 2.615392471 and -2.615392471.Subproblem 1
n + 2.416666667 = 2.615392471 Simplifying n + 2.416666667 = 2.615392471 Reorder the terms: 2.416666667 + n = 2.615392471 Solving 2.416666667 + n = 2.615392471 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + n = 2.615392471 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + n = 2.615392471 + -2.416666667 n = 2.615392471 + -2.416666667 Combine like terms: 2.615392471 + -2.416666667 = 0.198725804 n = 0.198725804 Simplifying n = 0.198725804Subproblem 2
n + 2.416666667 = -2.615392471 Simplifying n + 2.416666667 = -2.615392471 Reorder the terms: 2.416666667 + n = -2.615392471 Solving 2.416666667 + n = -2.615392471 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + n = -2.615392471 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + n = -2.615392471 + -2.416666667 n = -2.615392471 + -2.416666667 Combine like terms: -2.615392471 + -2.416666667 = -5.032059138 n = -5.032059138 Simplifying n = -5.032059138Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.198725804, -5.032059138}
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