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Simplifying 2n2 + -2n = 28 Reorder the terms: -2n + 2n2 = 28 Solving -2n + 2n2 = 28 Solving for variable 'n'. Reorder the terms: -28 + -2n + 2n2 = 28 + -28 Combine like terms: 28 + -28 = 0 -28 + -2n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-14 + -1n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-14 + -1n + n2)' equal to zero and attempt to solve: Simplifying -14 + -1n + n2 = 0 Solving -14 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '14' to each side of the equation. -14 + -1n + 14 + n2 = 0 + 14 Reorder the terms: -14 + 14 + -1n + n2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + -1n + n2 = 0 + 14 -1n + n2 = 0 + 14 Combine like terms: 0 + 14 = 14 -1n + n2 = 14 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 14 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 14 + 0.25 Combine like terms: 14 + 0.25 = 14.25 0.25 + -1n + n2 = 14.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (n + -0.5) equal to 3.774917218 and -3.774917218.Subproblem 1
n + -0.5 = 3.774917218 Simplifying n + -0.5 = 3.774917218 Reorder the terms: -0.5 + n = 3.774917218 Solving -0.5 + n = 3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 3.774917218 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 3.774917218 + 0.5 n = 3.774917218 + 0.5 Combine like terms: 3.774917218 + 0.5 = 4.274917218 n = 4.274917218 Simplifying n = 4.274917218Subproblem 2
n + -0.5 = -3.774917218 Simplifying n + -0.5 = -3.774917218 Reorder the terms: -0.5 + n = -3.774917218 Solving -0.5 + n = -3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -3.774917218 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -3.774917218 + 0.5 n = -3.774917218 + 0.5 Combine like terms: -3.774917218 + 0.5 = -3.274917218 n = -3.274917218 Simplifying n = -3.274917218Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.274917218, -3.274917218}Solution
n = {4.274917218, -3.274917218}
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