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Simplifying 6(12n + -6) + 8n(n + -10) = 0 Reorder the terms: 6(-6 + 12n) + 8n(n + -10) = 0 (-6 * 6 + 12n * 6) + 8n(n + -10) = 0 (-36 + 72n) + 8n(n + -10) = 0 Reorder the terms: -36 + 72n + 8n(-10 + n) = 0 -36 + 72n + (-10 * 8n + n * 8n) = 0 -36 + 72n + (-80n + 8n2) = 0 Combine like terms: 72n + -80n = -8n -36 + -8n + 8n2 = 0 Solving -36 + -8n + 8n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4'. 4(-9 + -2n + 2n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-9 + -2n + 2n2)' equal to zero and attempt to solve: Simplifying -9 + -2n + 2n2 = 0 Solving -9 + -2n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + -1n + n2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + -1n + 4.5 + n2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + -1n + n2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + -1n + n2 = 0 + 4.5 -1n + n2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 -1n + n2 = 4.5 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 = 4.5 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 4.5 + 0.25 Combine like terms: 4.5 + 0.25 = 4.75 0.25 + -1n + n2 = 4.75 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (n + -0.5) equal to 2.179449472 and -2.179449472.Subproblem 1
n + -0.5 = 2.179449472 Simplifying n + -0.5 = 2.179449472 Reorder the terms: -0.5 + n = 2.179449472 Solving -0.5 + n = 2.179449472 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 = 2.179449472 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 2.179449472 + 0.5 n = 2.179449472 + 0.5 Combine like terms: 2.179449472 + 0.5 = 2.679449472 n = 2.679449472 Simplifying n = 2.679449472Subproblem 2
n + -0.5 = -2.179449472 Simplifying n + -0.5 = -2.179449472 Reorder the terms: -0.5 + n = -2.179449472 Solving -0.5 + n = -2.179449472 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 = -2.179449472 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -2.179449472 + 0.5 n = -2.179449472 + 0.5 Combine like terms: -2.179449472 + 0.5 = -1.679449472 n = -1.679449472 Simplifying n = -1.679449472Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.679449472, -1.679449472}Solution
n = {2.679449472, -1.679449472}
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