(z^2)+12z+19=0

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Solution for (z^2)+12z+19=0 equation:


Simplifying
(z2) + 12z + 19 = 0
z2 + 12z + 19 = 0

Reorder the terms:
19 + 12z + z2 = 0

Solving
19 + 12z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + 12z + -19 + z2 = 0 + -19

Reorder the terms:
19 + -19 + 12z + z2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + 12z + z2 = 0 + -19
12z + z2 = 0 + -19

Combine like terms: 0 + -19 = -19
12z + z2 = -19

The z term is 12z.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12z + 36 + z2 = -19 + 36

Reorder the terms:
36 + 12z + z2 = -19 + 36

Combine like terms: -19 + 36 = 17
36 + 12z + z2 = 17

Factor a perfect square on the left side:
(z + 6)(z + 6) = 17

Calculate the square root of the right side: 4.123105626

Break this problem into two subproblems by setting 
(z + 6) equal to 4.123105626 and -4.123105626.

Subproblem 1

z + 6 = 4.123105626 Simplifying z + 6 = 4.123105626 Reorder the terms: 6 + z = 4.123105626 Solving 6 + z = 4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = 4.123105626 + -6 Combine like terms: 6 + -6 = 0 0 + z = 4.123105626 + -6 z = 4.123105626 + -6 Combine like terms: 4.123105626 + -6 = -1.876894374 z = -1.876894374 Simplifying z = -1.876894374

Subproblem 2

z + 6 = -4.123105626 Simplifying z + 6 = -4.123105626 Reorder the terms: 6 + z = -4.123105626 Solving 6 + z = -4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = -4.123105626 + -6 Combine like terms: 6 + -6 = 0 0 + z = -4.123105626 + -6 z = -4.123105626 + -6 Combine like terms: -4.123105626 + -6 = -10.123105626 z = -10.123105626 Simplifying z = -10.123105626

Solution

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

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