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Simplifying 16n2 + 52n + 3 = 0 Reorder the terms: 3 + 52n + 16n2 = 0 Solving 3 + 52n + 16n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + 3.25n + n2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + 3.25n + -0.1875 + n2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + 3.25n + n2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + 3.25n + n2 = 0 + -0.1875 3.25n + n2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 3.25n + n2 = -0.1875 The n term is 3.25n. Take half its coefficient (1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. 3.25n + 2.640625 + n2 = -0.1875 + 2.640625 Reorder the terms: 2.640625 + 3.25n + n2 = -0.1875 + 2.640625 Combine like terms: -0.1875 + 2.640625 = 2.453125 2.640625 + 3.25n + n2 = 2.453125 Factor a perfect square on the left side: (n + 1.625)(n + 1.625) = 2.453125 Calculate the square root of the right side: 1.566245511 Break this problem into two subproblems by setting (n + 1.625) equal to 1.566245511 and -1.566245511.Subproblem 1
n + 1.625 = 1.566245511 Simplifying n + 1.625 = 1.566245511 Reorder the terms: 1.625 + n = 1.566245511 Solving 1.625 + n = 1.566245511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + n = 1.566245511 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + n = 1.566245511 + -1.625 n = 1.566245511 + -1.625 Combine like terms: 1.566245511 + -1.625 = -0.058754489 n = -0.058754489 Simplifying n = -0.058754489Subproblem 2
n + 1.625 = -1.566245511 Simplifying n + 1.625 = -1.566245511 Reorder the terms: 1.625 + n = -1.566245511 Solving 1.625 + n = -1.566245511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + n = -1.566245511 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + n = -1.566245511 + -1.625 n = -1.566245511 + -1.625 Combine like terms: -1.566245511 + -1.625 = -3.191245511 n = -3.191245511 Simplifying n = -3.191245511Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.058754489, -3.191245511}
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