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Simplifying 4n2 + -17n = 19 Reorder the terms: -17n + 4n2 = 19 Solving -17n + 4n2 = 19 Solving for variable 'n'. Reorder the terms: -19 + -17n + 4n2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + -17n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -4.75 + -4.25n + n2 = 0 Move the constant term to the right: Add '4.75' to each side of the equation. -4.75 + -4.25n + 4.75 + n2 = 0 + 4.75 Reorder the terms: -4.75 + 4.75 + -4.25n + n2 = 0 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + -4.25n + n2 = 0 + 4.75 -4.25n + n2 = 0 + 4.75 Combine like terms: 0 + 4.75 = 4.75 -4.25n + n2 = 4.75 The n term is -4.25n. Take half its coefficient (-2.125). Square it (4.515625) and add it to both sides. Add '4.515625' to each side of the equation. -4.25n + 4.515625 + n2 = 4.75 + 4.515625 Reorder the terms: 4.515625 + -4.25n + n2 = 4.75 + 4.515625 Combine like terms: 4.75 + 4.515625 = 9.265625 4.515625 + -4.25n + n2 = 9.265625 Factor a perfect square on the left side: (n + -2.125)(n + -2.125) = 9.265625 Calculate the square root of the right side: 3.043948915 Break this problem into two subproblems by setting (n + -2.125) equal to 3.043948915 and -3.043948915.Subproblem 1
n + -2.125 = 3.043948915 Simplifying n + -2.125 = 3.043948915 Reorder the terms: -2.125 + n = 3.043948915 Solving -2.125 + n = 3.043948915 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + n = 3.043948915 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + n = 3.043948915 + 2.125 n = 3.043948915 + 2.125 Combine like terms: 3.043948915 + 2.125 = 5.168948915 n = 5.168948915 Simplifying n = 5.168948915Subproblem 2
n + -2.125 = -3.043948915 Simplifying n + -2.125 = -3.043948915 Reorder the terms: -2.125 + n = -3.043948915 Solving -2.125 + n = -3.043948915 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + n = -3.043948915 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + n = -3.043948915 + 2.125 n = -3.043948915 + 2.125 Combine like terms: -3.043948915 + 2.125 = -0.918948915 n = -0.918948915 Simplifying n = -0.918948915Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.168948915, -0.918948915}
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