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Simplifying n2 + 12n + 19 = 0 Reorder the terms: 19 + 12n + n2 = 0 Solving 19 + 12n + 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 + 12n + -19 + n2 = 0 + -19 Reorder the terms: 19 + -19 + 12n + n2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + 12n + n2 = 0 + -19 12n + n2 = 0 + -19 Combine like terms: 0 + -19 = -19 12n + n2 = -19 The n term is 12n. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12n + 36 + n2 = -19 + 36 Reorder the terms: 36 + 12n + n2 = -19 + 36 Combine like terms: -19 + 36 = 17 36 + 12n + n2 = 17 Factor a perfect square on the left side: (n + 6)(n + 6) = 17 Calculate the square root of the right side: 4.123105626 Break this problem into two subproblems by setting (n + 6) equal to 4.123105626 and -4.123105626.Subproblem 1
n + 6 = 4.123105626 Simplifying n + 6 = 4.123105626 Reorder the terms: 6 + n = 4.123105626 Solving 6 + n = 4.123105626 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + n = 4.123105626 + -6 Combine like terms: 6 + -6 = 0 0 + n = 4.123105626 + -6 n = 4.123105626 + -6 Combine like terms: 4.123105626 + -6 = -1.876894374 n = -1.876894374 Simplifying n = -1.876894374Subproblem 2
n + 6 = -4.123105626 Simplifying n + 6 = -4.123105626 Reorder the terms: 6 + n = -4.123105626 Solving 6 + n = -4.123105626 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + n = -4.123105626 + -6 Combine like terms: 6 + -6 = 0 0 + n = -4.123105626 + -6 n = -4.123105626 + -6 Combine like terms: -4.123105626 + -6 = -10.123105626 n = -10.123105626 Simplifying n = -10.123105626Solution
The solution to the problem is based on the solutions from the subproblems. n = {-1.876894374, -10.123105626}
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