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Simplifying 2n2 + 14n = 24 Reorder the terms: 14n + 2n2 = 24 Solving 14n + 2n2 = 24 Solving for variable 'n'. Reorder the terms: -24 + 14n + 2n2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + 14n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + 7n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + 7n + n2)' equal to zero and attempt to solve: Simplifying -12 + 7n + n2 = 0 Solving -12 + 7n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + 7n + 12 + n2 = 0 + 12 Reorder the terms: -12 + 12 + 7n + n2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 7n + n2 = 0 + 12 7n + n2 = 0 + 12 Combine like terms: 0 + 12 = 12 7n + n2 = 12 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 = 12 + 12.25 Reorder the terms: 12.25 + 7n + n2 = 12 + 12.25 Combine like terms: 12 + 12.25 = 24.25 12.25 + 7n + n2 = 24.25 Factor a perfect square on the left side: (n + 3.5)(n + 3.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (n + 3.5) equal to 4.924428901 and -4.924428901.Subproblem 1
n + 3.5 = 4.924428901 Simplifying n + 3.5 = 4.924428901 Reorder the terms: 3.5 + n = 4.924428901 Solving 3.5 + n = 4.924428901 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 = 4.924428901 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 4.924428901 + -3.5 n = 4.924428901 + -3.5 Combine like terms: 4.924428901 + -3.5 = 1.424428901 n = 1.424428901 Simplifying n = 1.424428901Subproblem 2
n + 3.5 = -4.924428901 Simplifying n + 3.5 = -4.924428901 Reorder the terms: 3.5 + n = -4.924428901 Solving 3.5 + n = -4.924428901 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 = -4.924428901 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -4.924428901 + -3.5 n = -4.924428901 + -3.5 Combine like terms: -4.924428901 + -3.5 = -8.424428901 n = -8.424428901 Simplifying n = -8.424428901Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.424428901, -8.424428901}Solution
n = {1.424428901, -8.424428901}
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