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Simplifying -4(2n + -6) * n = n + 6 Reorder the terms: -4(-6 + 2n) * n = n + 6 Reorder the terms for easier multiplication: -4n(-6 + 2n) = n + 6 (-6 * -4n + 2n * -4n) = n + 6 (24n + -8n2) = n + 6 Reorder the terms: 24n + -8n2 = 6 + n Solving 24n + -8n2 = 6 + n Solving for variable 'n'. Reorder the terms: -6 + 24n + -1n + -8n2 = 6 + n + -6 + -1n Combine like terms: 24n + -1n = 23n -6 + 23n + -8n2 = 6 + n + -6 + -1n Reorder the terms: -6 + 23n + -8n2 = 6 + -6 + n + -1n Combine like terms: 6 + -6 = 0 -6 + 23n + -8n2 = 0 + n + -1n -6 + 23n + -8n2 = n + -1n Combine like terms: n + -1n = 0 -6 + 23n + -8n2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. 0.75 + -2.875n + n2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -2.875n + -0.75 + n2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -2.875n + n2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -2.875n + n2 = 0 + -0.75 -2.875n + n2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -2.875n + n2 = -0.75 The n term is -2.875n. Take half its coefficient (-1.4375). Square it (2.06640625) and add it to both sides. Add '2.06640625' to each side of the equation. -2.875n + 2.06640625 + n2 = -0.75 + 2.06640625 Reorder the terms: 2.06640625 + -2.875n + n2 = -0.75 + 2.06640625 Combine like terms: -0.75 + 2.06640625 = 1.31640625 2.06640625 + -2.875n + n2 = 1.31640625 Factor a perfect square on the left side: (n + -1.4375)(n + -1.4375) = 1.31640625 Calculate the square root of the right side: 1.147347484 Break this problem into two subproblems by setting (n + -1.4375) equal to 1.147347484 and -1.147347484.Subproblem 1
n + -1.4375 = 1.147347484 Simplifying n + -1.4375 = 1.147347484 Reorder the terms: -1.4375 + n = 1.147347484 Solving -1.4375 + n = 1.147347484 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + n = 1.147347484 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + n = 1.147347484 + 1.4375 n = 1.147347484 + 1.4375 Combine like terms: 1.147347484 + 1.4375 = 2.584847484 n = 2.584847484 Simplifying n = 2.584847484Subproblem 2
n + -1.4375 = -1.147347484 Simplifying n + -1.4375 = -1.147347484 Reorder the terms: -1.4375 + n = -1.147347484 Solving -1.4375 + n = -1.147347484 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + n = -1.147347484 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + n = -1.147347484 + 1.4375 n = -1.147347484 + 1.4375 Combine like terms: -1.147347484 + 1.4375 = 0.290152516 n = 0.290152516 Simplifying n = 0.290152516Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.584847484, 0.290152516}
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