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Simplifying 2n2 + 14n + -12 = 0 Reorder the terms: -12 + 14n + 2n2 = 0 Solving -12 + 14n + 2n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-6 + 7n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-6 + 7n + n2)' equal to zero and attempt to solve: Simplifying -6 + 7n + n2 = 0 Solving -6 + 7n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '6' to each side of the equation. -6 + 7n + 6 + n2 = 0 + 6 Reorder the terms: -6 + 6 + 7n + n2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 7n + n2 = 0 + 6 7n + n2 = 0 + 6 Combine like terms: 0 + 6 = 6 7n + n2 = 6 The n term is 7n. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7n + 12.25 + n2 = 6 + 12.25 Reorder the terms: 12.25 + 7n + n2 = 6 + 12.25 Combine like terms: 6 + 12.25 = 18.25 12.25 + 7n + n2 = 18.25 Factor a perfect square on the left side: (n + 3.5)(n + 3.5) = 18.25 Calculate the square root of the right side: 4.272001873 Break this problem into two subproblems by setting (n + 3.5) equal to 4.272001873 and -4.272001873.Subproblem 1
n + 3.5 = 4.272001873 Simplifying n + 3.5 = 4.272001873 Reorder the terms: 3.5 + n = 4.272001873 Solving 3.5 + n = 4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 4.272001873 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 4.272001873 + -3.5 n = 4.272001873 + -3.5 Combine like terms: 4.272001873 + -3.5 = 0.772001873 n = 0.772001873 Simplifying n = 0.772001873Subproblem 2
n + 3.5 = -4.272001873 Simplifying n + 3.5 = -4.272001873 Reorder the terms: 3.5 + n = -4.272001873 Solving 3.5 + n = -4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -4.272001873 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -4.272001873 + -3.5 n = -4.272001873 + -3.5 Combine like terms: -4.272001873 + -3.5 = -7.772001873 n = -7.772001873 Simplifying n = -7.772001873Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.772001873, -7.772001873}Solution
n = {0.772001873, -7.772001873}
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