n^2+12n+19=0

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Solution for n^2+12n+19=0 equation:


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 = -1.876894374

Subproblem 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 = -10.123105626

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-1.876894374, -10.123105626}

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