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Simplifying 4n2 + 14n = 1 Reorder the terms: 14n + 4n2 = 1 Solving 14n + 4n2 = 1 Solving for variable 'n'. Reorder the terms: -1 + 14n + 4n2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 14n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + 3.5n + n2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 3.5n + 0.25 + n2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 3.5n + n2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 3.5n + n2 = 0 + 0.25 3.5n + n2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 3.5n + n2 = 0.25 The n term is 3.5n. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5n + 3.0625 + n2 = 0.25 + 3.0625 Reorder the terms: 3.0625 + 3.5n + n2 = 0.25 + 3.0625 Combine like terms: 0.25 + 3.0625 = 3.3125 3.0625 + 3.5n + n2 = 3.3125 Factor a perfect square on the left side: (n + 1.75)(n + 1.75) = 3.3125 Calculate the square root of the right side: 1.820027472 Break this problem into two subproblems by setting (n + 1.75) equal to 1.820027472 and -1.820027472.Subproblem 1
n + 1.75 = 1.820027472 Simplifying n + 1.75 = 1.820027472 Reorder the terms: 1.75 + n = 1.820027472 Solving 1.75 + n = 1.820027472 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = 1.820027472 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = 1.820027472 + -1.75 n = 1.820027472 + -1.75 Combine like terms: 1.820027472 + -1.75 = 0.070027472 n = 0.070027472 Simplifying n = 0.070027472Subproblem 2
n + 1.75 = -1.820027472 Simplifying n + 1.75 = -1.820027472 Reorder the terms: 1.75 + n = -1.820027472 Solving 1.75 + n = -1.820027472 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = -1.820027472 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = -1.820027472 + -1.75 n = -1.820027472 + -1.75 Combine like terms: -1.820027472 + -1.75 = -3.570027472 n = -3.570027472 Simplifying n = -3.570027472Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.070027472, -3.570027472}
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