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Simplifying 8n2 + -19n + -24 = 0 Reorder the terms: -24 + -19n + 8n2 = 0 Solving -24 + -19n + 8n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -3 + -2.375n + n2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -2.375n + 3 + n2 = 0 + 3 Reorder the terms: -3 + 3 + -2.375n + n2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -2.375n + n2 = 0 + 3 -2.375n + n2 = 0 + 3 Combine like terms: 0 + 3 = 3 -2.375n + n2 = 3 The n term is -2.375n. Take half its coefficient (-1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. -2.375n + 1.41015625 + n2 = 3 + 1.41015625 Reorder the terms: 1.41015625 + -2.375n + n2 = 3 + 1.41015625 Combine like terms: 3 + 1.41015625 = 4.41015625 1.41015625 + -2.375n + n2 = 4.41015625 Factor a perfect square on the left side: (n + -1.1875)(n + -1.1875) = 4.41015625 Calculate the square root of the right side: 2.100037202 Break this problem into two subproblems by setting (n + -1.1875) equal to 2.100037202 and -2.100037202.Subproblem 1
n + -1.1875 = 2.100037202 Simplifying n + -1.1875 = 2.100037202 Reorder the terms: -1.1875 + n = 2.100037202 Solving -1.1875 + n = 2.100037202 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + n = 2.100037202 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + n = 2.100037202 + 1.1875 n = 2.100037202 + 1.1875 Combine like terms: 2.100037202 + 1.1875 = 3.287537202 n = 3.287537202 Simplifying n = 3.287537202Subproblem 2
n + -1.1875 = -2.100037202 Simplifying n + -1.1875 = -2.100037202 Reorder the terms: -1.1875 + n = -2.100037202 Solving -1.1875 + n = -2.100037202 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + n = -2.100037202 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + n = -2.100037202 + 1.1875 n = -2.100037202 + 1.1875 Combine like terms: -2.100037202 + 1.1875 = -0.912537202 n = -0.912537202 Simplifying n = -0.912537202Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.287537202, -0.912537202}
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