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Simplifying 8n2 + 8 = 6 + -9n Reorder the terms: 8 + 8n2 = 6 + -9n Solving 8 + 8n2 = 6 + -9n Solving for variable 'n'. Reorder the terms: 8 + -6 + 9n + 8n2 = 6 + -9n + -6 + 9n Combine like terms: 8 + -6 = 2 2 + 9n + 8n2 = 6 + -9n + -6 + 9n Reorder the terms: 2 + 9n + 8n2 = 6 + -6 + -9n + 9n Combine like terms: 6 + -6 = 0 2 + 9n + 8n2 = 0 + -9n + 9n 2 + 9n + 8n2 = -9n + 9n Combine like terms: -9n + 9n = 0 2 + 9n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.25 + 1.125n + n2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + 1.125n + -0.25 + n2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + 1.125n + n2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + 1.125n + n2 = 0 + -0.25 1.125n + n2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 1.125n + n2 = -0.25 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 = -0.25 + 0.31640625 Reorder the terms: 0.31640625 + 1.125n + n2 = -0.25 + 0.31640625 Combine like terms: -0.25 + 0.31640625 = 0.06640625 0.31640625 + 1.125n + n2 = 0.06640625 Factor a perfect square on the left side: (n + 0.5625)(n + 0.5625) = 0.06640625 Calculate the square root of the right side: 0.257694102 Break this problem into two subproblems by setting (n + 0.5625) equal to 0.257694102 and -0.257694102.Subproblem 1
n + 0.5625 = 0.257694102 Simplifying n + 0.5625 = 0.257694102 Reorder the terms: 0.5625 + n = 0.257694102 Solving 0.5625 + n = 0.257694102 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 = 0.257694102 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = 0.257694102 + -0.5625 n = 0.257694102 + -0.5625 Combine like terms: 0.257694102 + -0.5625 = -0.304805898 n = -0.304805898 Simplifying n = -0.304805898Subproblem 2
n + 0.5625 = -0.257694102 Simplifying n + 0.5625 = -0.257694102 Reorder the terms: 0.5625 + n = -0.257694102 Solving 0.5625 + n = -0.257694102 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 = -0.257694102 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = -0.257694102 + -0.5625 n = -0.257694102 + -0.5625 Combine like terms: -0.257694102 + -0.5625 = -0.820194102 n = -0.820194102 Simplifying n = -0.820194102Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.304805898, -0.820194102}
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