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Simplifying 14 = 0.5(n + 6)(n + 6) + -12 Reorder the terms: 14 = 0.5(6 + n)(n + 6) + -12 Reorder the terms: 14 = 0.5(6 + n)(6 + n) + -12 Multiply (6 + n) * (6 + n) 14 = 0.5(6(6 + n) + n(6 + n)) + -12 14 = 0.5((6 * 6 + n * 6) + n(6 + n)) + -12 14 = 0.5((36 + 6n) + n(6 + n)) + -12 14 = 0.5(36 + 6n + (6 * n + n * n)) + -12 14 = 0.5(36 + 6n + (6n + n2)) + -12 Combine like terms: 6n + 6n = 12n 14 = 0.5(36 + 12n + n2) + -12 14 = (36 * 0.5 + 12n * 0.5 + n2 * 0.5) + -12 14 = (18 + 6n + 0.5n2) + -12 Reorder the terms: 14 = 18 + -12 + 6n + 0.5n2 Combine like terms: 18 + -12 = 6 14 = 6 + 6n + 0.5n2 Solving 14 = 6 + 6n + 0.5n2 Solving for variable 'n'. Combine like terms: 14 + -6 = 8 8 + -6n + -0.5n2 = 6 + 6n + 0.5n2 + -6 + -6n + -0.5n2 Reorder the terms: 8 + -6n + -0.5n2 = 6 + -6 + 6n + -6n + 0.5n2 + -0.5n2 Combine like terms: 6 + -6 = 0 8 + -6n + -0.5n2 = 0 + 6n + -6n + 0.5n2 + -0.5n2 8 + -6n + -0.5n2 = 6n + -6n + 0.5n2 + -0.5n2 Combine like terms: 6n + -6n = 0 8 + -6n + -0.5n2 = 0 + 0.5n2 + -0.5n2 8 + -6n + -0.5n2 = 0.5n2 + -0.5n2 Combine like terms: 0.5n2 + -0.5n2 = 0.0 8 + -6n + -0.5n2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -16 + 12n + n2 = 0 Move the constant term to the right: Add '16' to each side of the equation. -16 + 12n + 16 + n2 = 0 + 16 Reorder the terms: -16 + 16 + 12n + n2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 12n + n2 = 0 + 16 12n + n2 = 0 + 16 Combine like terms: 0 + 16 = 16 12n + n2 = 16 The n term is 12n. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12n + 36 + n2 = 16 + 36 Reorder the terms: 36 + 12n + n2 = 16 + 36 Combine like terms: 16 + 36 = 52 36 + 12n + n2 = 52 Factor a perfect square on the left side: (n + 6)(n + 6) = 52 Calculate the square root of the right side: 7.211102551 Break this problem into two subproblems by setting (n + 6) equal to 7.211102551 and -7.211102551.Subproblem 1
n + 6 = 7.211102551 Simplifying n + 6 = 7.211102551 Reorder the terms: 6 + n = 7.211102551 Solving 6 + n = 7.211102551 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + n = 7.211102551 + -6 Combine like terms: 6 + -6 = 0 0 + n = 7.211102551 + -6 n = 7.211102551 + -6 Combine like terms: 7.211102551 + -6 = 1.211102551 n = 1.211102551 Simplifying n = 1.211102551Subproblem 2
n + 6 = -7.211102551 Simplifying n + 6 = -7.211102551 Reorder the terms: 6 + n = -7.211102551 Solving 6 + n = -7.211102551 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + n = -7.211102551 + -6 Combine like terms: 6 + -6 = 0 0 + n = -7.211102551 + -6 n = -7.211102551 + -6 Combine like terms: -7.211102551 + -6 = -13.211102551 n = -13.211102551 Simplifying n = -13.211102551Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.211102551, -13.211102551}
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