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Simplifying 2(n + n2) = 45 (n * 2 + n2 * 2) = 45 (2n + 2n2) = 45 Solving 2n + 2n2 = 45 Solving for variable 'n'. Reorder the terms: -45 + 2n + 2n2 = 45 + -45 Combine like terms: 45 + -45 = 0 -45 + 2n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -22.5 + n + n2 = 0 Move the constant term to the right: Add '22.5' to each side of the equation. -22.5 + n + 22.5 + n2 = 0 + 22.5 Reorder the terms: -22.5 + 22.5 + n + n2 = 0 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + n + n2 = 0 + 22.5 n + n2 = 0 + 22.5 Combine like terms: 0 + 22.5 = 22.5 n + n2 = 22.5 The n term is n. 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. + 0.25 + n2 = 22.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 22.5 + 0.25 Combine like terms: 22.5 + 0.25 = 22.75 1.25 + n2 = 22.75 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 22.75 Calculate the square root of the right side: 4.769696007 Break this problem into two subproblems by setting (n + 0.5) equal to 4.769696007 and -4.769696007.Subproblem 1
n + 0.5 = 4.769696007 Simplifying n + 0.5 = 4.769696007 Reorder the terms: 0.5 + n = 4.769696007 Solving 0.5 + n = 4.769696007 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 = 4.769696007 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 4.769696007 + -0.5 n = 4.769696007 + -0.5 Combine like terms: 4.769696007 + -0.5 = 4.269696007 n = 4.269696007 Simplifying n = 4.269696007Subproblem 2
n + 0.5 = -4.769696007 Simplifying n + 0.5 = -4.769696007 Reorder the terms: 0.5 + n = -4.769696007 Solving 0.5 + n = -4.769696007 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 = -4.769696007 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -4.769696007 + -0.5 n = -4.769696007 + -0.5 Combine like terms: -4.769696007 + -0.5 = -5.269696007 n = -5.269696007 Simplifying n = -5.269696007Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.269696007, -5.269696007}
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