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Simplifying 4n2 + 6n + 1 = 0 Reorder the terms: 1 + 6n + 4n2 = 0 Solving 1 + 6n + 4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + 1.5n + n2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + 1.5n + -0.25 + n2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + 1.5n + n2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + 1.5n + n2 = 0 + -0.25 1.5n + n2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 1.5n + n2 = -0.25 The n term is 1.5n. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5n + 0.5625 + n2 = -0.25 + 0.5625 Reorder the terms: 0.5625 + 1.5n + n2 = -0.25 + 0.5625 Combine like terms: -0.25 + 0.5625 = 0.3125 0.5625 + 1.5n + n2 = 0.3125 Factor a perfect square on the left side: (n + 0.75)(n + 0.75) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (n + 0.75) equal to 0.559016994 and -0.559016994.Subproblem 1
n + 0.75 = 0.559016994 Simplifying n + 0.75 = 0.559016994 Reorder the terms: 0.75 + n = 0.559016994 Solving 0.75 + n = 0.559016994 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = 0.559016994 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 0.559016994 + -0.75 n = 0.559016994 + -0.75 Combine like terms: 0.559016994 + -0.75 = -0.190983006 n = -0.190983006 Simplifying n = -0.190983006Subproblem 2
n + 0.75 = -0.559016994 Simplifying n + 0.75 = -0.559016994 Reorder the terms: 0.75 + n = -0.559016994 Solving 0.75 + n = -0.559016994 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = -0.559016994 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -0.559016994 + -0.75 n = -0.559016994 + -0.75 Combine like terms: -0.559016994 + -0.75 = -1.309016994 n = -1.309016994 Simplifying n = -1.309016994Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.190983006, -1.309016994}
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