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Simplifying 5n2 = 1 + 5n Solving 5n2 = 1 + 5n Solving for variable 'n'. Reorder the terms: -1 + -5n + 5n2 = 1 + 5n + -1 + -5n Reorder the terms: -1 + -5n + 5n2 = 1 + -1 + 5n + -5n Combine like terms: 1 + -1 = 0 -1 + -5n + 5n2 = 0 + 5n + -5n -1 + -5n + 5n2 = 5n + -5n Combine like terms: 5n + -5n = 0 -1 + -5n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.2 + -1n + n2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -1n + 0.2 + n2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + -1n + n2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -1n + n2 = 0 + 0.2 -1n + n2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 -1n + n2 = 0.2 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 0.2 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 0.2 + 0.25 Combine like terms: 0.2 + 0.25 = 0.45 0.25 + -1n + n2 = 0.45 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 0.45 Calculate the square root of the right side: 0.670820393 Break this problem into two subproblems by setting (n + -0.5) equal to 0.670820393 and -0.670820393.Subproblem 1
n + -0.5 = 0.670820393 Simplifying n + -0.5 = 0.670820393 Reorder the terms: -0.5 + n = 0.670820393 Solving -0.5 + n = 0.670820393 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 0.670820393 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 0.670820393 + 0.5 n = 0.670820393 + 0.5 Combine like terms: 0.670820393 + 0.5 = 1.170820393 n = 1.170820393 Simplifying n = 1.170820393Subproblem 2
n + -0.5 = -0.670820393 Simplifying n + -0.5 = -0.670820393 Reorder the terms: -0.5 + n = -0.670820393 Solving -0.5 + n = -0.670820393 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -0.670820393 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -0.670820393 + 0.5 n = -0.670820393 + 0.5 Combine like terms: -0.670820393 + 0.5 = -0.170820393 n = -0.170820393 Simplifying n = -0.170820393Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.170820393, -0.170820393}
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