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Simplifying 7n2 + 4n + -80 = 0 Reorder the terms: -80 + 4n + 7n2 = 0 Solving -80 + 4n + 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'. -11.42857143 + 0.5714285714n + n2 = 0 Move the constant term to the right: Add '11.42857143' to each side of the equation. -11.42857143 + 0.5714285714n + 11.42857143 + n2 = 0 + 11.42857143 Reorder the terms: -11.42857143 + 11.42857143 + 0.5714285714n + n2 = 0 + 11.42857143 Combine like terms: -11.42857143 + 11.42857143 = 0.00000000 0.00000000 + 0.5714285714n + n2 = 0 + 11.42857143 0.5714285714n + n2 = 0 + 11.42857143 Combine like terms: 0 + 11.42857143 = 11.42857143 0.5714285714n + n2 = 11.42857143 The n term is 0.5714285714n. Take half its coefficient (0.2857142857). Square it (0.08163265305) and add it to both sides. Add '0.08163265305' to each side of the equation. 0.5714285714n + 0.08163265305 + n2 = 11.42857143 + 0.08163265305 Reorder the terms: 0.08163265305 + 0.5714285714n + n2 = 11.42857143 + 0.08163265305 Combine like terms: 11.42857143 + 0.08163265305 = 11.51020408305 0.08163265305 + 0.5714285714n + n2 = 11.51020408305 Factor a perfect square on the left side: (n + 0.2857142857)(n + 0.2857142857) = 11.51020408305 Calculate the square root of the right side: 3.392669168 Break this problem into two subproblems by setting (n + 0.2857142857) equal to 3.392669168 and -3.392669168.Subproblem 1
n + 0.2857142857 = 3.392669168 Simplifying n + 0.2857142857 = 3.392669168 Reorder the terms: 0.2857142857 + n = 3.392669168 Solving 0.2857142857 + n = 3.392669168 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2857142857' to each side of the equation. 0.2857142857 + -0.2857142857 + n = 3.392669168 + -0.2857142857 Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000 0.0000000000 + n = 3.392669168 + -0.2857142857 n = 3.392669168 + -0.2857142857 Combine like terms: 3.392669168 + -0.2857142857 = 3.1069548823 n = 3.1069548823 Simplifying n = 3.1069548823Subproblem 2
n + 0.2857142857 = -3.392669168 Simplifying n + 0.2857142857 = -3.392669168 Reorder the terms: 0.2857142857 + n = -3.392669168 Solving 0.2857142857 + n = -3.392669168 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2857142857' to each side of the equation. 0.2857142857 + -0.2857142857 + n = -3.392669168 + -0.2857142857 Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000 0.0000000000 + n = -3.392669168 + -0.2857142857 n = -3.392669168 + -0.2857142857 Combine like terms: -3.392669168 + -0.2857142857 = -3.6783834537 n = -3.6783834537 Simplifying n = -3.6783834537Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.1069548823, -3.6783834537}
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