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Simplifying -2n(8n + -4) = -4(1 + 3n) Reorder the terms: -2n(-4 + 8n) = -4(1 + 3n) (-4 * -2n + 8n * -2n) = -4(1 + 3n) (8n + -16n2) = -4(1 + 3n) 8n + -16n2 = (1 * -4 + 3n * -4) 8n + -16n2 = (-4 + -12n) Solving 8n + -16n2 = -4 + -12n Solving for variable 'n'. Reorder the terms: 4 + 8n + 12n + -16n2 = -4 + -12n + 4 + 12n Combine like terms: 8n + 12n = 20n 4 + 20n + -16n2 = -4 + -12n + 4 + 12n Reorder the terms: 4 + 20n + -16n2 = -4 + 4 + -12n + 12n Combine like terms: -4 + 4 = 0 4 + 20n + -16n2 = 0 + -12n + 12n 4 + 20n + -16n2 = -12n + 12n Combine like terms: -12n + 12n = 0 4 + 20n + -16n2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(1 + 5n + -4n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1 + 5n + -4n2)' equal to zero and attempt to solve: Simplifying 1 + 5n + -4n2 = 0 Solving 1 + 5n + -4n2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.25 + -1.25n + n2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1.25n + 0.25 + n2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1.25n + n2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1.25n + n2 = 0 + 0.25 -1.25n + n2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1.25n + n2 = 0.25 The n term is -1.25n. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25n + 0.390625 + n2 = 0.25 + 0.390625 Reorder the terms: 0.390625 + -1.25n + n2 = 0.25 + 0.390625 Combine like terms: 0.25 + 0.390625 = 0.640625 0.390625 + -1.25n + n2 = 0.640625 Factor a perfect square on the left side: (n + -0.625)(n + -0.625) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (n + -0.625) equal to 0.80039053 and -0.80039053.Subproblem 1
n + -0.625 = 0.80039053 Simplifying n + -0.625 = 0.80039053 Reorder the terms: -0.625 + n = 0.80039053 Solving -0.625 + n = 0.80039053 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = 0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = 0.80039053 + 0.625 n = 0.80039053 + 0.625 Combine like terms: 0.80039053 + 0.625 = 1.42539053 n = 1.42539053 Simplifying n = 1.42539053Subproblem 2
n + -0.625 = -0.80039053 Simplifying n + -0.625 = -0.80039053 Reorder the terms: -0.625 + n = -0.80039053 Solving -0.625 + n = -0.80039053 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = -0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = -0.80039053 + 0.625 n = -0.80039053 + 0.625 Combine like terms: -0.80039053 + 0.625 = -0.17539053 n = -0.17539053 Simplifying n = -0.17539053Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.42539053, -0.17539053}Solution
n = {1.42539053, -0.17539053}
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