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Simplifying n2 + -6n + -19 = 0 Reorder the terms: -19 + -6n + n2 = 0 Solving -19 + -6n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '19' to each side of the equation. -19 + -6n + 19 + n2 = 0 + 19 Reorder the terms: -19 + 19 + -6n + n2 = 0 + 19 Combine like terms: -19 + 19 = 0 0 + -6n + n2 = 0 + 19 -6n + n2 = 0 + 19 Combine like terms: 0 + 19 = 19 -6n + n2 = 19 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = 19 + 9 Reorder the terms: 9 + -6n + n2 = 19 + 9 Combine like terms: 19 + 9 = 28 9 + -6n + n2 = 28 Factor a perfect square on the left side: (n + -3)(n + -3) = 28 Calculate the square root of the right side: 5.291502622 Break this problem into two subproblems by setting (n + -3) equal to 5.291502622 and -5.291502622.Subproblem 1
n + -3 = 5.291502622 Simplifying n + -3 = 5.291502622 Reorder the terms: -3 + n = 5.291502622 Solving -3 + n = 5.291502622 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 5.291502622 + 3 Combine like terms: -3 + 3 = 0 0 + n = 5.291502622 + 3 n = 5.291502622 + 3 Combine like terms: 5.291502622 + 3 = 8.291502622 n = 8.291502622 Simplifying n = 8.291502622Subproblem 2
n + -3 = -5.291502622 Simplifying n + -3 = -5.291502622 Reorder the terms: -3 + n = -5.291502622 Solving -3 + n = -5.291502622 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -5.291502622 + 3 Combine like terms: -3 + 3 = 0 0 + n = -5.291502622 + 3 n = -5.291502622 + 3 Combine like terms: -5.291502622 + 3 = -2.291502622 n = -2.291502622 Simplifying n = -2.291502622Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.291502622, -2.291502622}
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