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Simplifying 5(n + 6) + -2n(3n * 1) = 0 Reorder the terms: 5(6 + n) + -2n(3n * 1) = 0 (6 * 5 + n * 5) + -2n(3n * 1) = 0 (30 + 5n) + -2n(3n * 1) = 0 Reorder the terms for easier multiplication: 30 + 5n + -2n(3 * 1n) = 0 Multiply 3 * 1 30 + 5n + -2n(3n) = 0 Remove parenthesis around (3n) 30 + 5n + -2n * 3n = 0 Reorder the terms for easier multiplication: 30 + 5n + -2 * 3n * n = 0 Multiply -2 * 3 30 + 5n + -6n * n = 0 Multiply n * n 30 + 5n + -6n2 = 0 Solving 30 + 5n + -6n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -5 + -0.8333333333n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.8333333333n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -0.8333333333n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.8333333333n + n2 = 0 + 5 -0.8333333333n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.8333333333n + n2 = 5 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 = 5 + 0.1736111111 Reorder the terms: 0.1736111111 + -0.8333333333n + n2 = 5 + 0.1736111111 Combine like terms: 5 + 0.1736111111 = 5.1736111111 0.1736111111 + -0.8333333333n + n2 = 5.1736111111 Factor a perfect square on the left side: (n + -0.4166666667)(n + -0.4166666667) = 5.1736111111 Calculate the square root of the right side: 2.274557344 Break this problem into two subproblems by setting (n + -0.4166666667) equal to 2.274557344 and -2.274557344.Subproblem 1
n + -0.4166666667 = 2.274557344 Simplifying n + -0.4166666667 = 2.274557344 Reorder the terms: -0.4166666667 + n = 2.274557344 Solving -0.4166666667 + n = 2.274557344 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 = 2.274557344 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + n = 2.274557344 + 0.4166666667 n = 2.274557344 + 0.4166666667 Combine like terms: 2.274557344 + 0.4166666667 = 2.6912240107 n = 2.6912240107 Simplifying n = 2.6912240107Subproblem 2
n + -0.4166666667 = -2.274557344 Simplifying n + -0.4166666667 = -2.274557344 Reorder the terms: -0.4166666667 + n = -2.274557344 Solving -0.4166666667 + n = -2.274557344 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 = -2.274557344 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + n = -2.274557344 + 0.4166666667 n = -2.274557344 + 0.4166666667 Combine like terms: -2.274557344 + 0.4166666667 = -1.8578906773 n = -1.8578906773 Simplifying n = -1.8578906773Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.6912240107, -1.8578906773}
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