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Simplifying 0 = 2n2 + 10n + 6 Reorder the terms: 0 = 6 + 10n + 2n2 Solving 0 = 6 + 10n + 2n2 Solving for variable 'n'. Combine like terms: 0 + -6 = -6 -6 + -10n + -2n2 = 6 + 10n + 2n2 + -6 + -10n + -2n2 Reorder the terms: -6 + -10n + -2n2 = 6 + -6 + 10n + -10n + 2n2 + -2n2 Combine like terms: 6 + -6 = 0 -6 + -10n + -2n2 = 0 + 10n + -10n + 2n2 + -2n2 -6 + -10n + -2n2 = 10n + -10n + 2n2 + -2n2 Combine like terms: 10n + -10n = 0 -6 + -10n + -2n2 = 0 + 2n2 + -2n2 -6 + -10n + -2n2 = 2n2 + -2n2 Combine like terms: 2n2 + -2n2 = 0 -6 + -10n + -2n2 = 0 Factor out the Greatest Common Factor (GCF), '-2'. -2(3 + 5n + n2) = 0 Ignore the factor -2.Subproblem 1
Set the factor '(3 + 5n + n2)' equal to zero and attempt to solve: Simplifying 3 + 5n + n2 = 0 Solving 3 + 5n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + 5n + -3 + n2 = 0 + -3 Reorder the terms: 3 + -3 + 5n + n2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 5n + n2 = 0 + -3 5n + n2 = 0 + -3 Combine like terms: 0 + -3 = -3 5n + n2 = -3 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 = -3 + 6.25 Reorder the terms: 6.25 + 5n + n2 = -3 + 6.25 Combine like terms: -3 + 6.25 = 3.25 6.25 + 5n + n2 = 3.25 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (n + 2.5) equal to 1.802775638 and -1.802775638.Subproblem 1
n + 2.5 = 1.802775638 Simplifying n + 2.5 = 1.802775638 Reorder the terms: 2.5 + n = 1.802775638 Solving 2.5 + n = 1.802775638 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 = 1.802775638 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 1.802775638 + -2.5 n = 1.802775638 + -2.5 Combine like terms: 1.802775638 + -2.5 = -0.697224362 n = -0.697224362 Simplifying n = -0.697224362Subproblem 2
n + 2.5 = -1.802775638 Simplifying n + 2.5 = -1.802775638 Reorder the terms: 2.5 + n = -1.802775638 Solving 2.5 + n = -1.802775638 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 = -1.802775638 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -1.802775638 + -2.5 n = -1.802775638 + -2.5 Combine like terms: -1.802775638 + -2.5 = -4.302775638 n = -4.302775638 Simplifying n = -4.302775638Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.697224362, -4.302775638}Solution
n = {-0.697224362, -4.302775638}
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