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Simplifying 840 = 3n2 + 4n Reorder the terms: 840 = 4n + 3n2 Solving 840 = 4n + 3n2 Solving for variable 'n'. Reorder the terms: 840 + -4n + -3n2 = 4n + -4n + 3n2 + -3n2 Combine like terms: 4n + -4n = 0 840 + -4n + -3n2 = 0 + 3n2 + -3n2 840 + -4n + -3n2 = 3n2 + -3n2 Combine like terms: 3n2 + -3n2 = 0 840 + -4n + -3n2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -280 + 1.333333333n + n2 = 0 Move the constant term to the right: Add '280' to each side of the equation. -280 + 1.333333333n + 280 + n2 = 0 + 280 Reorder the terms: -280 + 280 + 1.333333333n + n2 = 0 + 280 Combine like terms: -280 + 280 = 0 0 + 1.333333333n + n2 = 0 + 280 1.333333333n + n2 = 0 + 280 Combine like terms: 0 + 280 = 280 1.333333333n + n2 = 280 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 = 280 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333n + n2 = 280 + 0.4444444442 Combine like terms: 280 + 0.4444444442 = 280.4444444442 0.4444444442 + 1.333333333n + n2 = 280.4444444442 Factor a perfect square on the left side: (n + 0.6666666665)(n + 0.6666666665) = 280.4444444442 Calculate the square root of the right side: 16.746475583 Break this problem into two subproblems by setting (n + 0.6666666665) equal to 16.746475583 and -16.746475583.Subproblem 1
n + 0.6666666665 = 16.746475583 Simplifying n + 0.6666666665 = 16.746475583 Reorder the terms: 0.6666666665 + n = 16.746475583 Solving 0.6666666665 + n = 16.746475583 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 = 16.746475583 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + n = 16.746475583 + -0.6666666665 n = 16.746475583 + -0.6666666665 Combine like terms: 16.746475583 + -0.6666666665 = 16.0798089165 n = 16.0798089165 Simplifying n = 16.0798089165Subproblem 2
n + 0.6666666665 = -16.746475583 Simplifying n + 0.6666666665 = -16.746475583 Reorder the terms: 0.6666666665 + n = -16.746475583 Solving 0.6666666665 + n = -16.746475583 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 = -16.746475583 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + n = -16.746475583 + -0.6666666665 n = -16.746475583 + -0.6666666665 Combine like terms: -16.746475583 + -0.6666666665 = -17.4131422495 n = -17.4131422495 Simplifying n = -17.4131422495Solution
The solution to the problem is based on the solutions from the subproblems. n = {16.0798089165, -17.4131422495}
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