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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 = 9.719387758 8.576530615 + 5.857142857n + n2 = 9.719387758 Factor a perfect square on the left side: (n + 2.928571429)(n + 2.928571429) = 9.719387758 Calculate the square root of the right side: 3.117593264 Break this problem into two subproblems by setting (n + 2.928571429) equal to 3.117593264 and -3.117593264.Subproblem 1
n + 2.928571429 = 3.117593264 Simplifying n + 2.928571429 = 3.117593264 Reorder the terms: 2.928571429 + n = 3.117593264 Solving 2.928571429 + n = 3.117593264 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 = 3.117593264 + -2.928571429 Combine like terms: 2.928571429 + -2.928571429 = 0.000000000 0.000000000 + n = 3.117593264 + -2.928571429 n = 3.117593264 + -2.928571429 Combine like terms: 3.117593264 + -2.928571429 = 0.189021835 n = 0.189021835 Simplifying n = 0.189021835Subproblem 2
n + 2.928571429 = -3.117593264 Simplifying n + 2.928571429 = -3.117593264 Reorder the terms: 2.928571429 + n = -3.117593264 Solving 2.928571429 + n = -3.117593264 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 = -3.117593264 + -2.928571429 Combine like terms: 2.928571429 + -2.928571429 = 0.000000000 0.000000000 + n = -3.117593264 + -2.928571429 n = -3.117593264 + -2.928571429 Combine like terms: -3.117593264 + -2.928571429 = -6.046164693 n = -6.046164693 Simplifying n = -6.046164693Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.189021835, -6.046164693}
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