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Simplifying 8n2 + -4 = 9n Reorder the terms: -4 + 8n2 = 9n Solving -4 + 8n2 = 9n Solving for variable 'n'. Reorder the terms: -4 + -9n + 8n2 = 9n + -9n Combine like terms: 9n + -9n = 0 -4 + -9n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.5 + -1.125n + n2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.125n + 0.5 + n2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.125n + n2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.125n + n2 = 0 + 0.5 -1.125n + n2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.125n + n2 = 0.5 The n term is -1.125n. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125n + 0.31640625 + n2 = 0.5 + 0.31640625 Reorder the terms: 0.31640625 + -1.125n + n2 = 0.5 + 0.31640625 Combine like terms: 0.5 + 0.31640625 = 0.81640625 0.31640625 + -1.125n + n2 = 0.81640625 Factor a perfect square on the left side: (n + -0.5625)(n + -0.5625) = 0.81640625 Calculate the square root of the right side: 0.903552018 Break this problem into two subproblems by setting (n + -0.5625) equal to 0.903552018 and -0.903552018.Subproblem 1
n + -0.5625 = 0.903552018 Simplifying n + -0.5625 = 0.903552018 Reorder the terms: -0.5625 + n = 0.903552018 Solving -0.5625 + n = 0.903552018 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + n = 0.903552018 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = 0.903552018 + 0.5625 n = 0.903552018 + 0.5625 Combine like terms: 0.903552018 + 0.5625 = 1.466052018 n = 1.466052018 Simplifying n = 1.466052018Subproblem 2
n + -0.5625 = -0.903552018 Simplifying n + -0.5625 = -0.903552018 Reorder the terms: -0.5625 + n = -0.903552018 Solving -0.5625 + n = -0.903552018 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + n = -0.903552018 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = -0.903552018 + 0.5625 n = -0.903552018 + 0.5625 Combine like terms: -0.903552018 + 0.5625 = -0.341052018 n = -0.341052018 Simplifying n = -0.341052018Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.466052018, -0.341052018}
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