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Simplifying 5[n + -2] * 4[2n + 6] = 2 Reorder the terms: 5[-2 + n] * 4[2n + 6] = 2 Reorder the terms: 5[-2 + n] * 4[6 + 2n] = 2 Reorder the terms for easier multiplication: 5 * 4[-2 + n][6 + 2n] = 2 Multiply 5 * 4 20[-2 + n][6 + 2n] = 2 Multiply [-2 + n] * [6 + 2n] 20[-2[6 + 2n] + n[6 + 2n]] = 2 20[[6 * -2 + 2n * -2] + n[6 + 2n]] = 2 20[[-12 + -4n] + n[6 + 2n]] = 2 20[-12 + -4n + [6 * n + 2n * n]] = 2 20[-12 + -4n + [6n + 2n2]] = 2 Combine like terms: -4n + 6n = 2n 20[-12 + 2n + 2n2] = 2 [-12 * 20 + 2n * 20 + 2n2 * 20] = 2 [-240 + 40n + 40n2] = 2 Solving -240 + 40n + 40n2 = 2 Solving for variable 'n'. Reorder the terms: -240 + -2 + 40n + 40n2 = 2 + -2 Combine like terms: -240 + -2 = -242 -242 + 40n + 40n2 = 2 + -2 Combine like terms: 2 + -2 = 0 -242 + 40n + 40n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-121 + 20n + 20n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-121 + 20n + 20n2)' equal to zero and attempt to solve: Simplifying -121 + 20n + 20n2 = 0 Solving -121 + 20n + 20n2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -6.05 + n + n2 = 0.0 Move the constant term to the right: Add '6.05' to each side of the equation. -6.05 + n + 6.05 + n2 = 0.0 + 6.05 Reorder the terms: -6.05 + 6.05 + n + n2 = 0.0 + 6.05 Combine like terms: -6.05 + 6.05 = 0.00 0.00 + n + n2 = 0.0 + 6.05 n + n2 = 0.0 + 6.05 Combine like terms: 0.0 + 6.05 = 6.05 n + n2 = 6.05 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 = 6.05 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 6.05 + 0.25 Combine like terms: 6.05 + 0.25 = 6.3 1.25 + n2 = 6.3 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 6.3 Calculate the square root of the right side: 2.50998008 Break this problem into two subproblems by setting (n + 0.5) equal to 2.50998008 and -2.50998008.Subproblem 1
n + 0.5 = 2.50998008 Simplifying n + 0.5 = 2.50998008 Reorder the terms: 0.5 + n = 2.50998008 Solving 0.5 + n = 2.50998008 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 = 2.50998008 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 2.50998008 + -0.5 n = 2.50998008 + -0.5 Combine like terms: 2.50998008 + -0.5 = 2.00998008 n = 2.00998008 Simplifying n = 2.00998008Subproblem 2
n + 0.5 = -2.50998008 Simplifying n + 0.5 = -2.50998008 Reorder the terms: 0.5 + n = -2.50998008 Solving 0.5 + n = -2.50998008 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 = -2.50998008 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -2.50998008 + -0.5 n = -2.50998008 + -0.5 Combine like terms: -2.50998008 + -0.5 = -3.00998008 n = -3.00998008 Simplifying n = -3.00998008Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.00998008, -3.00998008}Solution
n = {2.00998008, -3.00998008}
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