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Simplifying 8n * 2 + -2(2n + -7)(2n + 6) = 11n Reorder the terms for easier multiplication: 8 * 2n + -2(2n + -7)(2n + 6) = 11n Multiply 8 * 2 16n + -2(2n + -7)(2n + 6) = 11n Reorder the terms: 16n + -2(-7 + 2n)(2n + 6) = 11n Reorder the terms: 16n + -2(-7 + 2n)(6 + 2n) = 11n Multiply (-7 + 2n) * (6 + 2n) 16n + -2(-7(6 + 2n) + 2n * (6 + 2n)) = 11n 16n + -2((6 * -7 + 2n * -7) + 2n * (6 + 2n)) = 11n 16n + -2((-42 + -14n) + 2n * (6 + 2n)) = 11n 16n + -2(-42 + -14n + (6 * 2n + 2n * 2n)) = 11n 16n + -2(-42 + -14n + (12n + 4n2)) = 11n Combine like terms: -14n + 12n = -2n 16n + -2(-42 + -2n + 4n2) = 11n 16n + (-42 * -2 + -2n * -2 + 4n2 * -2) = 11n 16n + (84 + 4n + -8n2) = 11n Reorder the terms: 84 + 16n + 4n + -8n2 = 11n Combine like terms: 16n + 4n = 20n 84 + 20n + -8n2 = 11n Solving 84 + 20n + -8n2 = 11n Solving for variable 'n'. Reorder the terms: 84 + 20n + -11n + -8n2 = 11n + -11n Combine like terms: 20n + -11n = 9n 84 + 9n + -8n2 = 11n + -11n Combine like terms: 11n + -11n = 0 84 + 9n + -8n2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -10.5 + -1.125n + n2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + -1.125n + 10.5 + n2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + -1.125n + n2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + -1.125n + n2 = 0 + 10.5 -1.125n + n2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 -1.125n + n2 = 10.5 The n term is -1.125n. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125n + 0.31640625 + n2 = 10.5 + 0.31640625 Reorder the terms: 0.31640625 + -1.125n + n2 = 10.5 + 0.31640625 Combine like terms: 10.5 + 0.31640625 = 10.81640625 0.31640625 + -1.125n + n2 = 10.81640625 Factor a perfect square on the left side: (n + -0.5625)(n + -0.5625) = 10.81640625 Calculate the square root of the right side: 3.288830529 Break this problem into two subproblems by setting (n + -0.5625) equal to 3.288830529 and -3.288830529.Subproblem 1
n + -0.5625 = 3.288830529 Simplifying n + -0.5625 = 3.288830529 Reorder the terms: -0.5625 + n = 3.288830529 Solving -0.5625 + n = 3.288830529 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + n = 3.288830529 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = 3.288830529 + 0.5625 n = 3.288830529 + 0.5625 Combine like terms: 3.288830529 + 0.5625 = 3.851330529 n = 3.851330529 Simplifying n = 3.851330529Subproblem 2
n + -0.5625 = -3.288830529 Simplifying n + -0.5625 = -3.288830529 Reorder the terms: -0.5625 + n = -3.288830529 Solving -0.5625 + n = -3.288830529 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + n = -3.288830529 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = -3.288830529 + 0.5625 n = -3.288830529 + 0.5625 Combine like terms: -3.288830529 + 0.5625 = -2.726330529 n = -2.726330529 Simplifying n = -2.726330529Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.851330529, -2.726330529}
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