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Simplifying 8n + -2(2n + -7)(2n + 6) = 11n Reorder the terms: 8n + -2(-7 + 2n)(2n + 6) = 11n Reorder the terms: 8n + -2(-7 + 2n)(6 + 2n) = 11n Multiply (-7 + 2n) * (6 + 2n) 8n + -2(-7(6 + 2n) + 2n * (6 + 2n)) = 11n 8n + -2((6 * -7 + 2n * -7) + 2n * (6 + 2n)) = 11n 8n + -2((-42 + -14n) + 2n * (6 + 2n)) = 11n 8n + -2(-42 + -14n + (6 * 2n + 2n * 2n)) = 11n 8n + -2(-42 + -14n + (12n + 4n2)) = 11n Combine like terms: -14n + 12n = -2n 8n + -2(-42 + -2n + 4n2) = 11n 8n + (-42 * -2 + -2n * -2 + 4n2 * -2) = 11n 8n + (84 + 4n + -8n2) = 11n Reorder the terms: 84 + 8n + 4n + -8n2 = 11n Combine like terms: 8n + 4n = 12n 84 + 12n + -8n2 = 11n Solving 84 + 12n + -8n2 = 11n Solving for variable 'n'. Reorder the terms: 84 + 12n + -11n + -8n2 = 11n + -11n Combine like terms: 12n + -11n = 1n 84 + 1n + -8n2 = 11n + -11n Combine like terms: 11n + -11n = 0 84 + 1n + -8n2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -10.5 + -0.125n + n2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + -0.125n + 10.5 + n2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + -0.125n + n2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + -0.125n + n2 = 0 + 10.5 -0.125n + n2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 -0.125n + n2 = 10.5 The n term is -0.125n. Take half its coefficient (-0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. -0.125n + 0.00390625 + n2 = 10.5 + 0.00390625 Reorder the terms: 0.00390625 + -0.125n + n2 = 10.5 + 0.00390625 Combine like terms: 10.5 + 0.00390625 = 10.50390625 0.00390625 + -0.125n + n2 = 10.50390625 Factor a perfect square on the left side: (n + -0.0625)(n + -0.0625) = 10.50390625 Calculate the square root of the right side: 3.240973041 Break this problem into two subproblems by setting (n + -0.0625) equal to 3.240973041 and -3.240973041.Subproblem 1
n + -0.0625 = 3.240973041 Simplifying n + -0.0625 = 3.240973041 Reorder the terms: -0.0625 + n = 3.240973041 Solving -0.0625 + n = 3.240973041 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + n = 3.240973041 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + n = 3.240973041 + 0.0625 n = 3.240973041 + 0.0625 Combine like terms: 3.240973041 + 0.0625 = 3.303473041 n = 3.303473041 Simplifying n = 3.303473041Subproblem 2
n + -0.0625 = -3.240973041 Simplifying n + -0.0625 = -3.240973041 Reorder the terms: -0.0625 + n = -3.240973041 Solving -0.0625 + n = -3.240973041 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + n = -3.240973041 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + n = -3.240973041 + 0.0625 n = -3.240973041 + 0.0625 Combine like terms: -3.240973041 + 0.0625 = -3.178473041 n = -3.178473041 Simplifying n = -3.178473041Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.303473041, -3.178473041}
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