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Simplifying 8n2 = 6 + -9n Solving 8n2 = 6 + -9n Solving for variable 'n'. Reorder the terms: -6 + 9n + 8n2 = 6 + -9n + -6 + 9n Reorder the terms: -6 + 9n + 8n2 = 6 + -6 + -9n + 9n Combine like terms: 6 + -6 = 0 -6 + 9n + 8n2 = 0 + -9n + 9n -6 + 9n + 8n2 = -9n + 9n Combine like terms: -9n + 9n = 0 -6 + 9n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.75 + 1.125n + n2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1.125n + 0.75 + n2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1.125n + n2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1.125n + n2 = 0 + 0.75 1.125n + n2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1.125n + n2 = 0.75 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 = 0.75 + 0.31640625 Reorder the terms: 0.31640625 + 1.125n + n2 = 0.75 + 0.31640625 Combine like terms: 0.75 + 0.31640625 = 1.06640625 0.31640625 + 1.125n + n2 = 1.06640625 Factor a perfect square on the left side: (n + 0.5625)(n + 0.5625) = 1.06640625 Calculate the square root of the right side: 1.032669478 Break this problem into two subproblems by setting (n + 0.5625) equal to 1.032669478 and -1.032669478.Subproblem 1
n + 0.5625 = 1.032669478 Simplifying n + 0.5625 = 1.032669478 Reorder the terms: 0.5625 + n = 1.032669478 Solving 0.5625 + n = 1.032669478 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 = 1.032669478 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = 1.032669478 + -0.5625 n = 1.032669478 + -0.5625 Combine like terms: 1.032669478 + -0.5625 = 0.470169478 n = 0.470169478 Simplifying n = 0.470169478Subproblem 2
n + 0.5625 = -1.032669478 Simplifying n + 0.5625 = -1.032669478 Reorder the terms: 0.5625 + n = -1.032669478 Solving 0.5625 + n = -1.032669478 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 = -1.032669478 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = -1.032669478 + -0.5625 n = -1.032669478 + -0.5625 Combine like terms: -1.032669478 + -0.5625 = -1.595169478 n = -1.595169478 Simplifying n = -1.595169478Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.470169478, -1.595169478}
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