z^2+22z+19=0

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Solution for z^2+22z+19=0 equation:


Simplifying
z2 + 22z + 19 = 0

Reorder the terms:
19 + 22z + z2 = 0

Solving
19 + 22z + 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 + 22z + -19 + z2 = 0 + -19

Reorder the terms:
19 + -19 + 22z + z2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + 22z + z2 = 0 + -19
22z + z2 = 0 + -19

Combine like terms: 0 + -19 = -19
22z + z2 = -19

The z term is 22z.  Take half its coefficient (11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
22z + 121 + z2 = -19 + 121

Reorder the terms:
121 + 22z + z2 = -19 + 121

Combine like terms: -19 + 121 = 102
121 + 22z + z2 = 102

Factor a perfect square on the left side:
(z + 11)(z + 11) = 102

Calculate the square root of the right side: 10.099504938

Break this problem into two subproblems by setting 
(z + 11) equal to 10.099504938 and -10.099504938.

Subproblem 1

z + 11 = 10.099504938 Simplifying z + 11 = 10.099504938 Reorder the terms: 11 + z = 10.099504938 Solving 11 + z = 10.099504938 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-11' to each side of the equation. 11 + -11 + z = 10.099504938 + -11 Combine like terms: 11 + -11 = 0 0 + z = 10.099504938 + -11 z = 10.099504938 + -11 Combine like terms: 10.099504938 + -11 = -0.900495062 z = -0.900495062 Simplifying z = -0.900495062

Subproblem 2

z + 11 = -10.099504938 Simplifying z + 11 = -10.099504938 Reorder the terms: 11 + z = -10.099504938 Solving 11 + z = -10.099504938 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-11' to each side of the equation. 11 + -11 + z = -10.099504938 + -11 Combine like terms: 11 + -11 = 0 0 + z = -10.099504938 + -11 z = -10.099504938 + -11 Combine like terms: -10.099504938 + -11 = -21.099504938 z = -21.099504938 Simplifying z = -21.099504938

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.900495062, -21.099504938}

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