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Simplifying 2n2 + 2n = 110 Reorder the terms: 2n + 2n2 = 110 Solving 2n + 2n2 = 110 Solving for variable 'n'. Reorder the terms: -110 + 2n + 2n2 = 110 + -110 Combine like terms: 110 + -110 = 0 -110 + 2n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-55 + n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-55 + n + n2)' equal to zero and attempt to solve: Simplifying -55 + n + n2 = 0 Solving -55 + n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '55' to each side of the equation. -55 + n + 55 + n2 = 0 + 55 Reorder the terms: -55 + 55 + n + n2 = 0 + 55 Combine like terms: -55 + 55 = 0 0 + n + n2 = 0 + 55 n + n2 = 0 + 55 Combine like terms: 0 + 55 = 55 n + n2 = 55 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. n + 0.25 + n2 = 55 + 0.25 Reorder the terms: 0.25 + n + n2 = 55 + 0.25 Combine like terms: 55 + 0.25 = 55.25 0.25 + n + n2 = 55.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 55.25 Calculate the square root of the right side: 7.433034374 Break this problem into two subproblems by setting (n + 0.5) equal to 7.433034374 and -7.433034374.Subproblem 1
n + 0.5 = 7.433034374 Simplifying n + 0.5 = 7.433034374 Reorder the terms: 0.5 + n = 7.433034374 Solving 0.5 + n = 7.433034374 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 = 7.433034374 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 7.433034374 + -0.5 n = 7.433034374 + -0.5 Combine like terms: 7.433034374 + -0.5 = 6.933034374 n = 6.933034374 Simplifying n = 6.933034374Subproblem 2
n + 0.5 = -7.433034374 Simplifying n + 0.5 = -7.433034374 Reorder the terms: 0.5 + n = -7.433034374 Solving 0.5 + n = -7.433034374 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 = -7.433034374 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -7.433034374 + -0.5 n = -7.433034374 + -0.5 Combine like terms: -7.433034374 + -0.5 = -7.933034374 n = -7.933034374 Simplifying n = -7.933034374Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.933034374, -7.933034374}Solution
n = {6.933034374, -7.933034374}
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