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Simplifying 7n2 + 2n + -37 = 0 Reorder the terms: -37 + 2n + 7n2 = 0 Solving -37 + 2n + 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'. -5.285714286 + 0.2857142857n + n2 = 0 Move the constant term to the right: Add '5.285714286' to each side of the equation. -5.285714286 + 0.2857142857n + 5.285714286 + n2 = 0 + 5.285714286 Reorder the terms: -5.285714286 + 5.285714286 + 0.2857142857n + n2 = 0 + 5.285714286 Combine like terms: -5.285714286 + 5.285714286 = 0.000000000 0.000000000 + 0.2857142857n + n2 = 0 + 5.285714286 0.2857142857n + n2 = 0 + 5.285714286 Combine like terms: 0 + 5.285714286 = 5.285714286 0.2857142857n + n2 = 5.285714286 The n term is 0.2857142857n. Take half its coefficient (0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. 0.2857142857n + 0.02040816328 + n2 = 5.285714286 + 0.02040816328 Reorder the terms: 0.02040816328 + 0.2857142857n + n2 = 5.285714286 + 0.02040816328 Combine like terms: 5.285714286 + 0.02040816328 = 5.30612244928 0.02040816328 + 0.2857142857n + n2 = 5.30612244928 Factor a perfect square on the left side: (n + 0.1428571429)(n + 0.1428571429) = 5.30612244928 Calculate the square root of the right side: 2.303502214 Break this problem into two subproblems by setting (n + 0.1428571429) equal to 2.303502214 and -2.303502214.Subproblem 1
n + 0.1428571429 = 2.303502214 Simplifying n + 0.1428571429 = 2.303502214 Reorder the terms: 0.1428571429 + n = 2.303502214 Solving 0.1428571429 + n = 2.303502214 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + n = 2.303502214 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + n = 2.303502214 + -0.1428571429 n = 2.303502214 + -0.1428571429 Combine like terms: 2.303502214 + -0.1428571429 = 2.1606450711 n = 2.1606450711 Simplifying n = 2.1606450711Subproblem 2
n + 0.1428571429 = -2.303502214 Simplifying n + 0.1428571429 = -2.303502214 Reorder the terms: 0.1428571429 + n = -2.303502214 Solving 0.1428571429 + n = -2.303502214 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + n = -2.303502214 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + n = -2.303502214 + -0.1428571429 n = -2.303502214 + -0.1428571429 Combine like terms: -2.303502214 + -0.1428571429 = -2.4463593569 n = -2.4463593569 Simplifying n = -2.4463593569Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.1606450711, -2.4463593569}
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