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Simplifying 5n + 2n * 3n = 66 Reorder the terms for easier multiplication: 5n + 2 * 3n * n = 66 Multiply 2 * 3 5n + 6n * n = 66 Multiply n * n 5n + 6n2 = 66 Solving 5n + 6n2 = 66 Solving for variable 'n'. Reorder the terms: -66 + 5n + 6n2 = 66 + -66 Combine like terms: 66 + -66 = 0 -66 + 5n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -11 + 0.8333333333n + n2 = 0 Move the constant term to the right: Add '11' to each side of the equation. -11 + 0.8333333333n + 11 + n2 = 0 + 11 Reorder the terms: -11 + 11 + 0.8333333333n + n2 = 0 + 11 Combine like terms: -11 + 11 = 0 0 + 0.8333333333n + n2 = 0 + 11 0.8333333333n + n2 = 0 + 11 Combine like terms: 0 + 11 = 11 0.8333333333n + n2 = 11 The n term is 0.8333333333n. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333n + 0.1736111111 + n2 = 11 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333n + n2 = 11 + 0.1736111111 Combine like terms: 11 + 0.1736111111 = 11.1736111111 0.1736111111 + 0.8333333333n + n2 = 11.1736111111 Factor a perfect square on the left side: (n + 0.4166666667)(n + 0.4166666667) = 11.1736111111 Calculate the square root of the right side: 3.342695187 Break this problem into two subproblems by setting (n + 0.4166666667) equal to 3.342695187 and -3.342695187.Subproblem 1
n + 0.4166666667 = 3.342695187 Simplifying n + 0.4166666667 = 3.342695187 Reorder the terms: 0.4166666667 + n = 3.342695187 Solving 0.4166666667 + n = 3.342695187 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + n = 3.342695187 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + n = 3.342695187 + -0.4166666667 n = 3.342695187 + -0.4166666667 Combine like terms: 3.342695187 + -0.4166666667 = 2.9260285203 n = 2.9260285203 Simplifying n = 2.9260285203Subproblem 2
n + 0.4166666667 = -3.342695187 Simplifying n + 0.4166666667 = -3.342695187 Reorder the terms: 0.4166666667 + n = -3.342695187 Solving 0.4166666667 + n = -3.342695187 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + n = -3.342695187 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + n = -3.342695187 + -0.4166666667 n = -3.342695187 + -0.4166666667 Combine like terms: -3.342695187 + -0.4166666667 = -3.7593618537 n = -3.7593618537 Simplifying n = -3.7593618537Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.9260285203, -3.7593618537}
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