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Simplifying 10n2 + -34 = 65n Reorder the terms: -34 + 10n2 = 65n Solving -34 + 10n2 = 65n Solving for variable 'n'. Reorder the terms: -34 + -65n + 10n2 = 65n + -65n Combine like terms: 65n + -65n = 0 -34 + -65n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -3.4 + -6.5n + n2 = 0 Move the constant term to the right: Add '3.4' to each side of the equation. -3.4 + -6.5n + 3.4 + n2 = 0 + 3.4 Reorder the terms: -3.4 + 3.4 + -6.5n + n2 = 0 + 3.4 Combine like terms: -3.4 + 3.4 = 0.0 0.0 + -6.5n + n2 = 0 + 3.4 -6.5n + n2 = 0 + 3.4 Combine like terms: 0 + 3.4 = 3.4 -6.5n + n2 = 3.4 The n term is -6.5n. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5n + 10.5625 + n2 = 3.4 + 10.5625 Reorder the terms: 10.5625 + -6.5n + n2 = 3.4 + 10.5625 Combine like terms: 3.4 + 10.5625 = 13.9625 10.5625 + -6.5n + n2 = 13.9625 Factor a perfect square on the left side: (n + -3.25)(n + -3.25) = 13.9625 Calculate the square root of the right side: 3.736642878 Break this problem into two subproblems by setting (n + -3.25) equal to 3.736642878 and -3.736642878.Subproblem 1
n + -3.25 = 3.736642878 Simplifying n + -3.25 = 3.736642878 Reorder the terms: -3.25 + n = 3.736642878 Solving -3.25 + n = 3.736642878 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + n = 3.736642878 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + n = 3.736642878 + 3.25 n = 3.736642878 + 3.25 Combine like terms: 3.736642878 + 3.25 = 6.986642878 n = 6.986642878 Simplifying n = 6.986642878Subproblem 2
n + -3.25 = -3.736642878 Simplifying n + -3.25 = -3.736642878 Reorder the terms: -3.25 + n = -3.736642878 Solving -3.25 + n = -3.736642878 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + n = -3.736642878 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + n = -3.736642878 + 3.25 n = -3.736642878 + 3.25 Combine like terms: -3.736642878 + 3.25 = -0.486642878 n = -0.486642878 Simplifying n = -0.486642878Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.986642878, -0.486642878}
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