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Simplifying 9n * 9n + 6n = 3 Reorder the terms for easier multiplication: 9 * 9n * n + 6n = 3 Multiply 9 * 9 81n * n + 6n = 3 Multiply n * n 81n2 + 6n = 3 Reorder the terms: 6n + 81n2 = 3 Solving 6n + 81n2 = 3 Solving for variable 'n'. Reorder the terms: -3 + 6n + 81n2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 6n + 81n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + 2n + 27n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + 2n + 27n2)' equal to zero and attempt to solve: Simplifying -1 + 2n + 27n2 = 0 Solving -1 + 2n + 27n2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.03703703704 + 0.07407407407n + n2 = 0 Move the constant term to the right: Add '0.03703703704' to each side of the equation. -0.03703703704 + 0.07407407407n + 0.03703703704 + n2 = 0 + 0.03703703704 Reorder the terms: -0.03703703704 + 0.03703703704 + 0.07407407407n + n2 = 0 + 0.03703703704 Combine like terms: -0.03703703704 + 0.03703703704 = 0.00000000000 0.00000000000 + 0.07407407407n + n2 = 0 + 0.03703703704 0.07407407407n + n2 = 0 + 0.03703703704 Combine like terms: 0 + 0.03703703704 = 0.03703703704 0.07407407407n + n2 = 0.03703703704 The n term is 0.07407407407n. Take half its coefficient (0.03703703704). Square it (0.001371742113) and add it to both sides. Add '0.001371742113' to each side of the equation. 0.07407407407n + 0.001371742113 + n2 = 0.03703703704 + 0.001371742113 Reorder the terms: 0.001371742113 + 0.07407407407n + n2 = 0.03703703704 + 0.001371742113 Combine like terms: 0.03703703704 + 0.001371742113 = 0.038408779153 0.001371742113 + 0.07407407407n + n2 = 0.038408779153 Factor a perfect square on the left side: (n + 0.03703703704)(n + 0.03703703704) = 0.038408779153 Calculate the square root of the right side: 0.195981579 Break this problem into two subproblems by setting (n + 0.03703703704) equal to 0.195981579 and -0.195981579.Subproblem 1
n + 0.03703703704 = 0.195981579 Simplifying n + 0.03703703704 = 0.195981579 Reorder the terms: 0.03703703704 + n = 0.195981579 Solving 0.03703703704 + n = 0.195981579 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.03703703704' to each side of the equation. 0.03703703704 + -0.03703703704 + n = 0.195981579 + -0.03703703704 Combine like terms: 0.03703703704 + -0.03703703704 = 0.00000000000 0.00000000000 + n = 0.195981579 + -0.03703703704 n = 0.195981579 + -0.03703703704 Combine like terms: 0.195981579 + -0.03703703704 = 0.15894454196 n = 0.15894454196 Simplifying n = 0.15894454196Subproblem 2
n + 0.03703703704 = -0.195981579 Simplifying n + 0.03703703704 = -0.195981579 Reorder the terms: 0.03703703704 + n = -0.195981579 Solving 0.03703703704 + n = -0.195981579 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.03703703704' to each side of the equation. 0.03703703704 + -0.03703703704 + n = -0.195981579 + -0.03703703704 Combine like terms: 0.03703703704 + -0.03703703704 = 0.00000000000 0.00000000000 + n = -0.195981579 + -0.03703703704 n = -0.195981579 + -0.03703703704 Combine like terms: -0.195981579 + -0.03703703704 = -0.23301861604 n = -0.23301861604 Simplifying n = -0.23301861604Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.15894454196, -0.23301861604}Solution
n = {0.15894454196, -0.23301861604}
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