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Simplifying (2n)(2n) + n + -120 = 0 Remove parenthesis around (2n) 2n(2n) + n + -120 = 0 Remove parenthesis around (2n) 2n * 2n + n + -120 = 0 Reorder the terms for easier multiplication: 2 * 2n * n + n + -120 = 0 Multiply 2 * 2 4n * n + n + -120 = 0 Multiply n * n 4n2 + n + -120 = 0 Reorder the terms: -120 + n + 4n2 = 0 Solving -120 + n + 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'. -30 + 0.25n + n2 = 0 Move the constant term to the right: Add '30' to each side of the equation. -30 + 0.25n + 30 + n2 = 0 + 30 Reorder the terms: -30 + 30 + 0.25n + n2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + 0.25n + n2 = 0 + 30 0.25n + n2 = 0 + 30 Combine like terms: 0 + 30 = 30 0.25n + n2 = 30 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. 0.25n + 0.25 + n2 = 30 + 0.25 Reorder the terms: 0.25 + 0.25n + n2 = 30 + 0.25 Combine like terms: 30 + 0.25 = 30.25 0.25 + 0.25n + n2 = 30.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 30.25 Calculate the square root of the right side: 5.5 Break this problem into two subproblems by setting (n + 0.5) equal to 5.5 and -5.5.Subproblem 1
n + 0.5 = 5.5 Simplifying n + 0.5 = 5.5 Reorder the terms: 0.5 + n = 5.5 Solving 0.5 + n = 5.5 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 = 5.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 5.5 + -0.5 n = 5.5 + -0.5 Combine like terms: 5.5 + -0.5 = 5 n = 5 Simplifying n = 5Subproblem 2
n + 0.5 = -5.5 Simplifying n + 0.5 = -5.5 Reorder the terms: 0.5 + n = -5.5 Solving 0.5 + n = -5.5 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 = -5.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -5.5 + -0.5 n = -5.5 + -0.5 Combine like terms: -5.5 + -0.5 = -6 n = -6 Simplifying n = -6Solution
The solution to the problem is based on the solutions from the subproblems. n = {5, -6}
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