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Simplifying -22n(-23n + 1) = 45 Reorder the terms: -22n(1 + -23n) = 45 (1 * -22n + -23n * -22n) = 45 (-22n + 506n2) = 45 Solving -22n + 506n2 = 45 Solving for variable 'n'. Reorder the terms: -45 + -22n + 506n2 = 45 + -45 Combine like terms: 45 + -45 = 0 -45 + -22n + 506n2 = 0 Begin completing the square. Divide all terms by 506 the coefficient of the squared term: Divide each side by '506'. -0.08893280632 + -0.04347826087n + n2 = 0 Move the constant term to the right: Add '0.08893280632' to each side of the equation. -0.08893280632 + -0.04347826087n + 0.08893280632 + n2 = 0 + 0.08893280632 Reorder the terms: -0.08893280632 + 0.08893280632 + -0.04347826087n + n2 = 0 + 0.08893280632 Combine like terms: -0.08893280632 + 0.08893280632 = 0.00000000000 0.00000000000 + -0.04347826087n + n2 = 0 + 0.08893280632 -0.04347826087n + n2 = 0 + 0.08893280632 Combine like terms: 0 + 0.08893280632 = 0.08893280632 -0.04347826087n + n2 = 0.08893280632 The n term is -0.04347826087n. Take half its coefficient (-0.02173913044). Square it (0.0004725897923) and add it to both sides. Add '0.0004725897923' to each side of the equation. -0.04347826087n + 0.0004725897923 + n2 = 0.08893280632 + 0.0004725897923 Reorder the terms: 0.0004725897923 + -0.04347826087n + n2 = 0.08893280632 + 0.0004725897923 Combine like terms: 0.08893280632 + 0.0004725897923 = 0.0894053961123 0.0004725897923 + -0.04347826087n + n2 = 0.0894053961123 Factor a perfect square on the left side: (n + -0.02173913044)(n + -0.02173913044) = 0.0894053961123 Calculate the square root of the right side: 0.299007351 Break this problem into two subproblems by setting (n + -0.02173913044) equal to 0.299007351 and -0.299007351.Subproblem 1
n + -0.02173913044 = 0.299007351 Simplifying n + -0.02173913044 = 0.299007351 Reorder the terms: -0.02173913044 + n = 0.299007351 Solving -0.02173913044 + n = 0.299007351 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.02173913044' to each side of the equation. -0.02173913044 + 0.02173913044 + n = 0.299007351 + 0.02173913044 Combine like terms: -0.02173913044 + 0.02173913044 = 0.00000000000 0.00000000000 + n = 0.299007351 + 0.02173913044 n = 0.299007351 + 0.02173913044 Combine like terms: 0.299007351 + 0.02173913044 = 0.32074648144 n = 0.32074648144 Simplifying n = 0.32074648144Subproblem 2
n + -0.02173913044 = -0.299007351 Simplifying n + -0.02173913044 = -0.299007351 Reorder the terms: -0.02173913044 + n = -0.299007351 Solving -0.02173913044 + n = -0.299007351 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.02173913044' to each side of the equation. -0.02173913044 + 0.02173913044 + n = -0.299007351 + 0.02173913044 Combine like terms: -0.02173913044 + 0.02173913044 = 0.00000000000 0.00000000000 + n = -0.299007351 + 0.02173913044 n = -0.299007351 + 0.02173913044 Combine like terms: -0.299007351 + 0.02173913044 = -0.27726822056 n = -0.27726822056 Simplifying n = -0.27726822056Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.32074648144, -0.27726822056}
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