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Simplifying 2n2 + -10n = 24 Reorder the terms: -10n + 2n2 = 24 Solving -10n + 2n2 = 24 Solving for variable 'n'. Reorder the terms: -24 + -10n + 2n2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + -10n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + -5n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + -5n + n2)' equal to zero and attempt to solve: Simplifying -12 + -5n + n2 = 0 Solving -12 + -5n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + -5n + 12 + n2 = 0 + 12 Reorder the terms: -12 + 12 + -5n + n2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + -5n + n2 = 0 + 12 -5n + n2 = 0 + 12 Combine like terms: 0 + 12 = 12 -5n + n2 = 12 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 = 12 + 6.25 Reorder the terms: 6.25 + -5n + n2 = 12 + 6.25 Combine like terms: 12 + 6.25 = 18.25 6.25 + -5n + n2 = 18.25 Factor a perfect square on the left side: (n + -2.5)(n + -2.5) = 18.25 Calculate the square root of the right side: 4.272001873 Break this problem into two subproblems by setting (n + -2.5) equal to 4.272001873 and -4.272001873.Subproblem 1
n + -2.5 = 4.272001873 Simplifying n + -2.5 = 4.272001873 Reorder the terms: -2.5 + n = 4.272001873 Solving -2.5 + n = 4.272001873 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 = 4.272001873 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 4.272001873 + 2.5 n = 4.272001873 + 2.5 Combine like terms: 4.272001873 + 2.5 = 6.772001873 n = 6.772001873 Simplifying n = 6.772001873Subproblem 2
n + -2.5 = -4.272001873 Simplifying n + -2.5 = -4.272001873 Reorder the terms: -2.5 + n = -4.272001873 Solving -2.5 + n = -4.272001873 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 = -4.272001873 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -4.272001873 + 2.5 n = -4.272001873 + 2.5 Combine like terms: -4.272001873 + 2.5 = -1.772001873 n = -1.772001873 Simplifying n = -1.772001873Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.772001873, -1.772001873}Solution
n = {6.772001873, -1.772001873}
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