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Simplifying -5n + -8n(1 + 7n) = -8 -5n + (1 * -8n + 7n * -8n) = -8 -5n + (-8n + -56n2) = -8 Combine like terms: -5n + -8n = -13n -13n + -56n2 = -8 Solving -13n + -56n2 = -8 Solving for variable 'n'. Reorder the terms: 8 + -13n + -56n2 = -8 + 8 Combine like terms: -8 + 8 = 0 8 + -13n + -56n2 = 0 Begin completing the square. Divide all terms by -56 the coefficient of the squared term: Divide each side by '-56'. -0.1428571429 + 0.2321428571n + n2 = 0 Move the constant term to the right: Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.2321428571n + 0.1428571429 + n2 = 0 + 0.1428571429 Reorder the terms: -0.1428571429 + 0.1428571429 + 0.2321428571n + n2 = 0 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + 0.2321428571n + n2 = 0 + 0.1428571429 0.2321428571n + n2 = 0 + 0.1428571429 Combine like terms: 0 + 0.1428571429 = 0.1428571429 0.2321428571n + n2 = 0.1428571429 The n term is 0.2321428571n. Take half its coefficient (0.1160714286). Square it (0.01347257654) and add it to both sides. Add '0.01347257654' to each side of the equation. 0.2321428571n + 0.01347257654 + n2 = 0.1428571429 + 0.01347257654 Reorder the terms: 0.01347257654 + 0.2321428571n + n2 = 0.1428571429 + 0.01347257654 Combine like terms: 0.1428571429 + 0.01347257654 = 0.15632971944 0.01347257654 + 0.2321428571n + n2 = 0.15632971944 Factor a perfect square on the left side: (n + 0.1160714286)(n + 0.1160714286) = 0.15632971944 Calculate the square root of the right side: 0.395385533 Break this problem into two subproblems by setting (n + 0.1160714286) equal to 0.395385533 and -0.395385533.Subproblem 1
n + 0.1160714286 = 0.395385533 Simplifying n + 0.1160714286 = 0.395385533 Reorder the terms: 0.1160714286 + n = 0.395385533 Solving 0.1160714286 + n = 0.395385533 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1160714286' to each side of the equation. 0.1160714286 + -0.1160714286 + n = 0.395385533 + -0.1160714286 Combine like terms: 0.1160714286 + -0.1160714286 = 0.0000000000 0.0000000000 + n = 0.395385533 + -0.1160714286 n = 0.395385533 + -0.1160714286 Combine like terms: 0.395385533 + -0.1160714286 = 0.2793141044 n = 0.2793141044 Simplifying n = 0.2793141044Subproblem 2
n + 0.1160714286 = -0.395385533 Simplifying n + 0.1160714286 = -0.395385533 Reorder the terms: 0.1160714286 + n = -0.395385533 Solving 0.1160714286 + n = -0.395385533 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1160714286' to each side of the equation. 0.1160714286 + -0.1160714286 + n = -0.395385533 + -0.1160714286 Combine like terms: 0.1160714286 + -0.1160714286 = 0.0000000000 0.0000000000 + n = -0.395385533 + -0.1160714286 n = -0.395385533 + -0.1160714286 Combine like terms: -0.395385533 + -0.1160714286 = -0.5114569616 n = -0.5114569616 Simplifying n = -0.5114569616Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.2793141044, -0.5114569616}
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