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Simplifying -4n2 + -2n + 1 = 0 Reorder the terms: 1 + -2n + -4n2 = 0 Solving 1 + -2n + -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 + 0.5n + n2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 0.5n + 0.25 + n2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 0.5n + n2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 0.5n + n2 = 0 + 0.25 0.5n + n2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 0.5n + n2 = 0.25 The n term is 0.5n. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5n + 0.0625 + n2 = 0.25 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 0.25 + 0.0625 Combine like terms: 0.25 + 0.0625 = 0.3125 0.0625 + 0.5n + n2 = 0.3125 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (n + 0.25) equal to 0.559016994 and -0.559016994.Subproblem 1
n + 0.25 = 0.559016994 Simplifying n + 0.25 = 0.559016994 Reorder the terms: 0.25 + n = 0.559016994 Solving 0.25 + n = 0.559016994 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 0.559016994 + -0.25 n = 0.559016994 + -0.25 Combine like terms: 0.559016994 + -0.25 = 0.309016994 n = 0.309016994 Simplifying n = 0.309016994Subproblem 2
n + 0.25 = -0.559016994 Simplifying n + 0.25 = -0.559016994 Reorder the terms: 0.25 + n = -0.559016994 Solving 0.25 + n = -0.559016994 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -0.559016994 + -0.25 n = -0.559016994 + -0.25 Combine like terms: -0.559016994 + -0.25 = -0.809016994 n = -0.809016994 Simplifying n = -0.809016994Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.309016994, -0.809016994}
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