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Simplifying 3n2 + 4n = 9 Reorder the terms: 4n + 3n2 = 9 Solving 4n + 3n2 = 9 Solving for variable 'n'. Reorder the terms: -9 + 4n + 3n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + 4n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3 + 1.333333333n + n2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 1.333333333n + 3 + n2 = 0 + 3 Reorder the terms: -3 + 3 + 1.333333333n + n2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 1.333333333n + n2 = 0 + 3 1.333333333n + n2 = 0 + 3 Combine like terms: 0 + 3 = 3 1.333333333n + n2 = 3 The n term is 1.333333333n. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333n + 0.4444444442 + n2 = 3 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333n + n2 = 3 + 0.4444444442 Combine like terms: 3 + 0.4444444442 = 3.4444444442 0.4444444442 + 1.333333333n + n2 = 3.4444444442 Factor a perfect square on the left side: (n + 0.6666666665)(n + 0.6666666665) = 3.4444444442 Calculate the square root of the right side: 1.855921454 Break this problem into two subproblems by setting (n + 0.6666666665) equal to 1.855921454 and -1.855921454.Subproblem 1
n + 0.6666666665 = 1.855921454 Simplifying n + 0.6666666665 = 1.855921454 Reorder the terms: 0.6666666665 + n = 1.855921454 Solving 0.6666666665 + n = 1.855921454 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + n = 1.855921454 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + n = 1.855921454 + -0.6666666665 n = 1.855921454 + -0.6666666665 Combine like terms: 1.855921454 + -0.6666666665 = 1.1892547875 n = 1.1892547875 Simplifying n = 1.1892547875Subproblem 2
n + 0.6666666665 = -1.855921454 Simplifying n + 0.6666666665 = -1.855921454 Reorder the terms: 0.6666666665 + n = -1.855921454 Solving 0.6666666665 + n = -1.855921454 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + n = -1.855921454 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + n = -1.855921454 + -0.6666666665 n = -1.855921454 + -0.6666666665 Combine like terms: -1.855921454 + -0.6666666665 = -2.5225881205 n = -2.5225881205 Simplifying n = -2.5225881205Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.1892547875, -2.5225881205}
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