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Simplifying 4n2 + 19n + -18 = 0 Reorder the terms: -18 + 19n + 4n2 = 0 Solving -18 + 19n + 4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -4.5 + 4.75n + n2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 4.75n + 4.5 + n2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 4.75n + n2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 4.75n + n2 = 0 + 4.5 4.75n + n2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 4.75n + n2 = 4.5 The n term is 4.75n. Take half its coefficient (2.375). Square it (5.640625) and add it to both sides. Add '5.640625' to each side of the equation. 4.75n + 5.640625 + n2 = 4.5 + 5.640625 Reorder the terms: 5.640625 + 4.75n + n2 = 4.5 + 5.640625 Combine like terms: 4.5 + 5.640625 = 10.140625 5.640625 + 4.75n + n2 = 10.140625 Factor a perfect square on the left side: (n + 2.375)(n + 2.375) = 10.140625 Calculate the square root of the right side: 3.184434801 Break this problem into two subproblems by setting (n + 2.375) equal to 3.184434801 and -3.184434801.Subproblem 1
n + 2.375 = 3.184434801 Simplifying n + 2.375 = 3.184434801 Reorder the terms: 2.375 + n = 3.184434801 Solving 2.375 + n = 3.184434801 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + n = 3.184434801 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + n = 3.184434801 + -2.375 n = 3.184434801 + -2.375 Combine like terms: 3.184434801 + -2.375 = 0.809434801 n = 0.809434801 Simplifying n = 0.809434801Subproblem 2
n + 2.375 = -3.184434801 Simplifying n + 2.375 = -3.184434801 Reorder the terms: 2.375 + n = -3.184434801 Solving 2.375 + n = -3.184434801 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + n = -3.184434801 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + n = -3.184434801 + -2.375 n = -3.184434801 + -2.375 Combine like terms: -3.184434801 + -2.375 = -5.559434801 n = -5.559434801 Simplifying n = -5.559434801Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.809434801, -5.559434801}
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