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Simplifying 7n2 + -41n + 8 = 0 Reorder the terms: 8 + -41n + 7n2 = 0 Solving 8 + -41n + 7n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 1.142857143 + -5.857142857n + n2 = 0 Move the constant term to the right: Add '-1.142857143' to each side of the equation. 1.142857143 + -5.857142857n + -1.142857143 + n2 = 0 + -1.142857143 Reorder the terms: 1.142857143 + -1.142857143 + -5.857142857n + n2 = 0 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + -5.857142857n + n2 = 0 + -1.142857143 -5.857142857n + n2 = 0 + -1.142857143 Combine like terms: 0 + -1.142857143 = -1.142857143 -5.857142857n + n2 = -1.142857143 The n term is -5.857142857n. Take half its coefficient (-2.928571429). Square it (8.576530615) and add it to both sides. Add '8.576530615' to each side of the equation. -5.857142857n + 8.576530615 + n2 = -1.142857143 + 8.576530615 Reorder the terms: 8.576530615 + -5.857142857n + n2 = -1.142857143 + 8.576530615 Combine like terms: -1.142857143 + 8.576530615 = 7.433673472 8.576530615 + -5.857142857n + n2 = 7.433673472 Factor a perfect square on the left side: (n + -2.928571429)(n + -2.928571429) = 7.433673472 Calculate the square root of the right side: 2.726476384 Break this problem into two subproblems by setting (n + -2.928571429) equal to 2.726476384 and -2.726476384.Subproblem 1
n + -2.928571429 = 2.726476384 Simplifying n + -2.928571429 = 2.726476384 Reorder the terms: -2.928571429 + n = 2.726476384 Solving -2.928571429 + n = 2.726476384 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.928571429' to each side of the equation. -2.928571429 + 2.928571429 + n = 2.726476384 + 2.928571429 Combine like terms: -2.928571429 + 2.928571429 = 0.000000000 0.000000000 + n = 2.726476384 + 2.928571429 n = 2.726476384 + 2.928571429 Combine like terms: 2.726476384 + 2.928571429 = 5.655047813 n = 5.655047813 Simplifying n = 5.655047813Subproblem 2
n + -2.928571429 = -2.726476384 Simplifying n + -2.928571429 = -2.726476384 Reorder the terms: -2.928571429 + n = -2.726476384 Solving -2.928571429 + n = -2.726476384 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.928571429' to each side of the equation. -2.928571429 + 2.928571429 + n = -2.726476384 + 2.928571429 Combine like terms: -2.928571429 + 2.928571429 = 0.000000000 0.000000000 + n = -2.726476384 + 2.928571429 n = -2.726476384 + 2.928571429 Combine like terms: -2.726476384 + 2.928571429 = 0.202095045 n = 0.202095045 Simplifying n = 0.202095045Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.655047813, 0.202095045}
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