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Simplifying 7n2 + 22n + 5 = 0 Reorder the terms: 5 + 22n + 7n2 = 0 Solving 5 + 22n + 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'. 0.7142857143 + 3.142857143n + n2 = 0 Move the constant term to the right: Add '-0.7142857143' to each side of the equation. 0.7142857143 + 3.142857143n + -0.7142857143 + n2 = 0 + -0.7142857143 Reorder the terms: 0.7142857143 + -0.7142857143 + 3.142857143n + n2 = 0 + -0.7142857143 Combine like terms: 0.7142857143 + -0.7142857143 = 0.0000000000 0.0000000000 + 3.142857143n + n2 = 0 + -0.7142857143 3.142857143n + n2 = 0 + -0.7142857143 Combine like terms: 0 + -0.7142857143 = -0.7142857143 3.142857143n + n2 = -0.7142857143 The n term is 3.142857143n. Take half its coefficient (1.571428572). Square it (2.469387757) and add it to both sides. Add '2.469387757' to each side of the equation. 3.142857143n + 2.469387757 + n2 = -0.7142857143 + 2.469387757 Reorder the terms: 2.469387757 + 3.142857143n + n2 = -0.7142857143 + 2.469387757 Combine like terms: -0.7142857143 + 2.469387757 = 1.7551020427 2.469387757 + 3.142857143n + n2 = 1.7551020427 Factor a perfect square on the left side: (n + 1.571428572)(n + 1.571428572) = 1.7551020427 Calculate the square root of the right side: 1.324802643 Break this problem into two subproblems by setting (n + 1.571428572) equal to 1.324802643 and -1.324802643.Subproblem 1
n + 1.571428572 = 1.324802643 Simplifying n + 1.571428572 = 1.324802643 Reorder the terms: 1.571428572 + n = 1.324802643 Solving 1.571428572 + n = 1.324802643 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.571428572' to each side of the equation. 1.571428572 + -1.571428572 + n = 1.324802643 + -1.571428572 Combine like terms: 1.571428572 + -1.571428572 = 0.000000000 0.000000000 + n = 1.324802643 + -1.571428572 n = 1.324802643 + -1.571428572 Combine like terms: 1.324802643 + -1.571428572 = -0.246625929 n = -0.246625929 Simplifying n = -0.246625929Subproblem 2
n + 1.571428572 = -1.324802643 Simplifying n + 1.571428572 = -1.324802643 Reorder the terms: 1.571428572 + n = -1.324802643 Solving 1.571428572 + n = -1.324802643 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.571428572' to each side of the equation. 1.571428572 + -1.571428572 + n = -1.324802643 + -1.571428572 Combine like terms: 1.571428572 + -1.571428572 = 0.000000000 0.000000000 + n = -1.324802643 + -1.571428572 n = -1.324802643 + -1.571428572 Combine like terms: -1.324802643 + -1.571428572 = -2.896231215 n = -2.896231215 Simplifying n = -2.896231215Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.246625929, -2.896231215}
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