5z^2+19z-4=1

Solution for 5z^2+19z-4=1 equation:

Simplifying
5z2 + 19z + -4 = 1

Reorder the terms:
-4 + 19z + 5z2 = 1

Solving
-4 + 19z + 5z2 = 1

Solving for variable 'z'.

Reorder the terms:
-4 + -1 + 19z + 5z2 = 1 + -1

Combine like terms: -4 + -1 = -5
-5 + 19z + 5z2 = 1 + -1

Combine like terms: 1 + -1 = 0
-5 + 19z + 5z2 = 0

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
-1 + 3.8z + z2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 3.8z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + 3.8z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 3.8z + z2 = 0 + 1
3.8z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
3.8z + z2 = 1

The z term is 3.8z.  Take half its coefficient (1.9).
Square it (3.61) and add it to both sides.

Add '3.61' to each side of the equation.
3.8z + 3.61 + z2 = 1 + 3.61

Reorder the terms:
3.61 + 3.8z + z2 = 1 + 3.61

Combine like terms: 1 + 3.61 = 4.61
3.61 + 3.8z + z2 = 4.61

Factor a perfect square on the left side:
(z + 1.9)(z + 1.9) = 4.61

Calculate the square root of the right side: 2.147091055

Break this problem into two subproblems by setting 
(z + 1.9) equal to 2.147091055 and -2.147091055.

Subproblem 1
z + 1.9 = 2.147091055

Simplifying
z + 1.9 = 2.147091055

Reorder the terms:
1.9 + z = 2.147091055

Solving
1.9 + z = 2.147091055

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-1.9' to each side of the equation.
1.9 + -1.9 + z = 2.147091055 + -1.9

Combine like terms: 1.9 + -1.9 = 0.0
0.0 + z = 2.147091055 + -1.9
z = 2.147091055 + -1.9

Combine like terms: 2.147091055 + -1.9 = 0.247091055
z = 0.247091055

Simplifying
z = 0.247091055

Subproblem 2
z + 1.9 = -2.147091055

Simplifying
z + 1.9 = -2.147091055

Reorder the terms:
1.9 + z = -2.147091055

Solving
1.9 + z = -2.147091055

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-1.9' to each side of the equation.
1.9 + -1.9 + z = -2.147091055 + -1.9

Combine like terms: 1.9 + -1.9 = 0.0
0.0 + z = -2.147091055 + -1.9
z = -2.147091055 + -1.9

Combine like terms: -2.147091055 + -1.9 = -4.047091055
z = -4.047091055

Simplifying
z = -4.047091055

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.247091055, -4.047091055}