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Simplifying 2n2 + 2 = 10n Reorder the terms: 2 + 2n2 = 10n Solving 2 + 2n2 = 10n Solving for variable 'n'. Reorder the terms: 2 + -10n + 2n2 = 10n + -10n Combine like terms: 10n + -10n = 0 2 + -10n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(1 + -5n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + -5n + n2)' equal to zero and attempt to solve: Simplifying 1 + -5n + n2 = 0 Solving 1 + -5n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -5n + -1 + n2 = 0 + -1 Reorder the terms: 1 + -1 + -5n + n2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -5n + n2 = 0 + -1 -5n + n2 = 0 + -1 Combine like terms: 0 + -1 = -1 -5n + n2 = -1 The n term is -5n. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5n + 6.25 + n2 = -1 + 6.25 Reorder the terms: 6.25 + -5n + n2 = -1 + 6.25 Combine like terms: -1 + 6.25 = 5.25 6.25 + -5n + n2 = 5.25 Factor a perfect square on the left side: (n + -2.5)(n + -2.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (n + -2.5) equal to 2.291287847 and -2.291287847.Subproblem 1
n + -2.5 = 2.291287847 Simplifying n + -2.5 = 2.291287847 Reorder the terms: -2.5 + n = 2.291287847 Solving -2.5 + n = 2.291287847 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = 2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 2.291287847 + 2.5 n = 2.291287847 + 2.5 Combine like terms: 2.291287847 + 2.5 = 4.791287847 n = 4.791287847 Simplifying n = 4.791287847Subproblem 2
n + -2.5 = -2.291287847 Simplifying n + -2.5 = -2.291287847 Reorder the terms: -2.5 + n = -2.291287847 Solving -2.5 + n = -2.291287847 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = -2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -2.291287847 + 2.5 n = -2.291287847 + 2.5 Combine like terms: -2.291287847 + 2.5 = 0.208712153 n = 0.208712153 Simplifying n = 0.208712153Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.791287847, 0.208712153}Solution
n = {4.791287847, 0.208712153}
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