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Simplifying 12n2 + 13n + -455 = 0 Reorder the terms: -455 + 13n + 12n2 = 0 Solving -455 + 13n + 12n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -37.91666667 + 1.083333333n + n2 = 0 Move the constant term to the right: Add '37.91666667' to each side of the equation. -37.91666667 + 1.083333333n + 37.91666667 + n2 = 0 + 37.91666667 Reorder the terms: -37.91666667 + 37.91666667 + 1.083333333n + n2 = 0 + 37.91666667 Combine like terms: -37.91666667 + 37.91666667 = 0.00000000 0.00000000 + 1.083333333n + n2 = 0 + 37.91666667 1.083333333n + n2 = 0 + 37.91666667 Combine like terms: 0 + 37.91666667 = 37.91666667 1.083333333n + n2 = 37.91666667 The n term is 1.083333333n. Take half its coefficient (0.5416666665). Square it (0.2934027776) and add it to both sides. Add '0.2934027776' to each side of the equation. 1.083333333n + 0.2934027776 + n2 = 37.91666667 + 0.2934027776 Reorder the terms: 0.2934027776 + 1.083333333n + n2 = 37.91666667 + 0.2934027776 Combine like terms: 37.91666667 + 0.2934027776 = 38.2100694476 0.2934027776 + 1.083333333n + n2 = 38.2100694476 Factor a perfect square on the left side: (n + 0.5416666665)(n + 0.5416666665) = 38.2100694476 Calculate the square root of the right side: 6.181429402 Break this problem into two subproblems by setting (n + 0.5416666665) equal to 6.181429402 and -6.181429402.Subproblem 1
n + 0.5416666665 = 6.181429402 Simplifying n + 0.5416666665 = 6.181429402 Reorder the terms: 0.5416666665 + n = 6.181429402 Solving 0.5416666665 + n = 6.181429402 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5416666665' to each side of the equation. 0.5416666665 + -0.5416666665 + n = 6.181429402 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + n = 6.181429402 + -0.5416666665 n = 6.181429402 + -0.5416666665 Combine like terms: 6.181429402 + -0.5416666665 = 5.6397627355 n = 5.6397627355 Simplifying n = 5.6397627355Subproblem 2
n + 0.5416666665 = -6.181429402 Simplifying n + 0.5416666665 = -6.181429402 Reorder the terms: 0.5416666665 + n = -6.181429402 Solving 0.5416666665 + n = -6.181429402 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5416666665' to each side of the equation. 0.5416666665 + -0.5416666665 + n = -6.181429402 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + n = -6.181429402 + -0.5416666665 n = -6.181429402 + -0.5416666665 Combine like terms: -6.181429402 + -0.5416666665 = -6.7230960685 n = -6.7230960685 Simplifying n = -6.7230960685Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.6397627355, -6.7230960685}
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