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Simplifying 9n2 + 4n + -16 = 0 Reorder the terms: -16 + 4n + 9n2 = 0 Solving -16 + 4n + 9n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.777777778 + 0.4444444444n + n2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + 0.4444444444n + 1.777777778 + n2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + 0.4444444444n + n2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + 0.4444444444n + n2 = 0 + 1.777777778 0.4444444444n + n2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 0.4444444444n + n2 = 1.777777778 The n term is 0.4444444444n. Take half its coefficient (0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. 0.4444444444n + 0.04938271604 + n2 = 1.777777778 + 0.04938271604 Reorder the terms: 0.04938271604 + 0.4444444444n + n2 = 1.777777778 + 0.04938271604 Combine like terms: 1.777777778 + 0.04938271604 = 1.82716049404 0.04938271604 + 0.4444444444n + n2 = 1.82716049404 Factor a perfect square on the left side: (n + 0.2222222222)(n + 0.2222222222) = 1.82716049404 Calculate the square root of the right side: 1.351725007 Break this problem into two subproblems by setting (n + 0.2222222222) equal to 1.351725007 and -1.351725007.Subproblem 1
n + 0.2222222222 = 1.351725007 Simplifying n + 0.2222222222 = 1.351725007 Reorder the terms: 0.2222222222 + n = 1.351725007 Solving 0.2222222222 + n = 1.351725007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2222222222' to each side of the equation. 0.2222222222 + -0.2222222222 + n = 1.351725007 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + n = 1.351725007 + -0.2222222222 n = 1.351725007 + -0.2222222222 Combine like terms: 1.351725007 + -0.2222222222 = 1.1295027848 n = 1.1295027848 Simplifying n = 1.1295027848Subproblem 2
n + 0.2222222222 = -1.351725007 Simplifying n + 0.2222222222 = -1.351725007 Reorder the terms: 0.2222222222 + n = -1.351725007 Solving 0.2222222222 + n = -1.351725007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2222222222' to each side of the equation. 0.2222222222 + -0.2222222222 + n = -1.351725007 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + n = -1.351725007 + -0.2222222222 n = -1.351725007 + -0.2222222222 Combine like terms: -1.351725007 + -0.2222222222 = -1.5739472292 n = -1.5739472292 Simplifying n = -1.5739472292Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.1295027848, -1.5739472292}
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