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Simplifying -8n2 + -3n + 15 = -8 Reorder the terms: 15 + -3n + -8n2 = -8 Solving 15 + -3n + -8n2 = -8 Solving for variable 'n'. Reorder the terms: 15 + 8 + -3n + -8n2 = -8 + 8 Combine like terms: 15 + 8 = 23 23 + -3n + -8n2 = -8 + 8 Combine like terms: -8 + 8 = 0 23 + -3n + -8n2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -2.875 + 0.375n + n2 = 0 Move the constant term to the right: Add '2.875' to each side of the equation. -2.875 + 0.375n + 2.875 + n2 = 0 + 2.875 Reorder the terms: -2.875 + 2.875 + 0.375n + n2 = 0 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + 0.375n + n2 = 0 + 2.875 0.375n + n2 = 0 + 2.875 Combine like terms: 0 + 2.875 = 2.875 0.375n + n2 = 2.875 The n term is 0.375n. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375n + 0.03515625 + n2 = 2.875 + 0.03515625 Reorder the terms: 0.03515625 + 0.375n + n2 = 2.875 + 0.03515625 Combine like terms: 2.875 + 0.03515625 = 2.91015625 0.03515625 + 0.375n + n2 = 2.91015625 Factor a perfect square on the left side: (n + 0.1875)(n + 0.1875) = 2.91015625 Calculate the square root of the right side: 1.705918008 Break this problem into two subproblems by setting (n + 0.1875) equal to 1.705918008 and -1.705918008.Subproblem 1
n + 0.1875 = 1.705918008 Simplifying n + 0.1875 = 1.705918008 Reorder the terms: 0.1875 + n = 1.705918008 Solving 0.1875 + n = 1.705918008 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + n = 1.705918008 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + n = 1.705918008 + -0.1875 n = 1.705918008 + -0.1875 Combine like terms: 1.705918008 + -0.1875 = 1.518418008 n = 1.518418008 Simplifying n = 1.518418008Subproblem 2
n + 0.1875 = -1.705918008 Simplifying n + 0.1875 = -1.705918008 Reorder the terms: 0.1875 + n = -1.705918008 Solving 0.1875 + n = -1.705918008 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + n = -1.705918008 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + n = -1.705918008 + -0.1875 n = -1.705918008 + -0.1875 Combine like terms: -1.705918008 + -0.1875 = -1.893418008 n = -1.893418008 Simplifying n = -1.893418008Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.518418008, -1.893418008}
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