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Simplifying 0 = 4n2 + -6n + -19 Reorder the terms: 0 = -19 + -6n + 4n2 Solving 0 = -19 + -6n + 4n2 Solving for variable 'n'. Combine like terms: 0 + 19 = 19 19 + 6n + -4n2 = -19 + -6n + 4n2 + 19 + 6n + -4n2 Reorder the terms: 19 + 6n + -4n2 = -19 + 19 + -6n + 6n + 4n2 + -4n2 Combine like terms: -19 + 19 = 0 19 + 6n + -4n2 = 0 + -6n + 6n + 4n2 + -4n2 19 + 6n + -4n2 = -6n + 6n + 4n2 + -4n2 Combine like terms: -6n + 6n = 0 19 + 6n + -4n2 = 0 + 4n2 + -4n2 19 + 6n + -4n2 = 4n2 + -4n2 Combine like terms: 4n2 + -4n2 = 0 19 + 6n + -4n2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -4.75 + -1.5n + n2 = 0 Move the constant term to the right: Add '4.75' to each side of the equation. -4.75 + -1.5n + 4.75 + n2 = 0 + 4.75 Reorder the terms: -4.75 + 4.75 + -1.5n + n2 = 0 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + -1.5n + n2 = 0 + 4.75 -1.5n + n2 = 0 + 4.75 Combine like terms: 0 + 4.75 = 4.75 -1.5n + n2 = 4.75 The n term is -1.5n. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5n + 0.5625 + n2 = 4.75 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 4.75 + 0.5625 Combine like terms: 4.75 + 0.5625 = 5.3125 0.5625 + -1.5n + n2 = 5.3125 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 5.3125 Calculate the square root of the right side: 2.304886114 Break this problem into two subproblems by setting (n + -0.75) equal to 2.304886114 and -2.304886114.Subproblem 1
n + -0.75 = 2.304886114 Simplifying n + -0.75 = 2.304886114 Reorder the terms: -0.75 + n = 2.304886114 Solving -0.75 + n = 2.304886114 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 2.304886114 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 2.304886114 + 0.75 n = 2.304886114 + 0.75 Combine like terms: 2.304886114 + 0.75 = 3.054886114 n = 3.054886114 Simplifying n = 3.054886114Subproblem 2
n + -0.75 = -2.304886114 Simplifying n + -0.75 = -2.304886114 Reorder the terms: -0.75 + n = -2.304886114 Solving -0.75 + n = -2.304886114 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -2.304886114 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -2.304886114 + 0.75 n = -2.304886114 + 0.75 Combine like terms: -2.304886114 + 0.75 = -1.554886114 n = -1.554886114 Simplifying n = -1.554886114Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.054886114, -1.554886114}
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