6z^2=17z-3

Solution for 6z^2=17z-3 equation:

Simplifying
6z2 = 17z + -3

Reorder the terms:
6z2 = -3 + 17z

Solving
6z2 = -3 + 17z

Solving for variable 'z'.

Reorder the terms:
3 + -17z + 6z2 = -3 + 17z + 3 + -17z

Reorder the terms:
3 + -17z + 6z2 = -3 + 3 + 17z + -17z

Combine like terms: -3 + 3 = 0
3 + -17z + 6z2 = 0 + 17z + -17z
3 + -17z + 6z2 = 17z + -17z

Combine like terms: 17z + -17z = 0
3 + -17z + 6z2 = 0

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

Divide each side by '6'.
0.5 + -2.833333333z + z2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2.833333333z + -0.5 + z2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2.833333333z + z2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2.833333333z + z2 = 0 + -0.5
-2.833333333z + z2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2.833333333z + z2 = -0.5

The z term is -2.833333333z.  Take half its coefficient (-1.416666667).
Square it (2.006944445) and add it to both sides.

Add '2.006944445' to each side of the equation.
-2.833333333z + 2.006944445 + z2 = -0.5 + 2.006944445

Reorder the terms:
2.006944445 + -2.833333333z + z2 = -0.5 + 2.006944445

Combine like terms: -0.5 + 2.006944445 = 1.506944445
2.006944445 + -2.833333333z + z2 = 1.506944445

Factor a perfect square on the left side:
(z + -1.416666667)(z + -1.416666667) = 1.506944445

Calculate the square root of the right side: 1.227576655

Break this problem into two subproblems by setting 
(z + -1.416666667) equal to 1.227576655 and -1.227576655.

Subproblem 1
z + -1.416666667 = 1.227576655

Simplifying
z + -1.416666667 = 1.227576655

Reorder the terms:
-1.416666667 + z = 1.227576655

Solving
-1.416666667 + z = 1.227576655

Solving for variable 'z'.

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

Add '1.416666667' to each side of the equation.
-1.416666667 + 1.416666667 + z = 1.227576655 + 1.416666667

Combine like terms: -1.416666667 + 1.416666667 = 0.000000000
0.000000000 + z = 1.227576655 + 1.416666667
z = 1.227576655 + 1.416666667

Combine like terms: 1.227576655 + 1.416666667 = 2.644243322
z = 2.644243322

Simplifying
z = 2.644243322

Subproblem 2
z + -1.416666667 = -1.227576655

Simplifying
z + -1.416666667 = -1.227576655

Reorder the terms:
-1.416666667 + z = -1.227576655

Solving
-1.416666667 + z = -1.227576655

Solving for variable 'z'.

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

Add '1.416666667' to each side of the equation.
-1.416666667 + 1.416666667 + z = -1.227576655 + 1.416666667

Combine like terms: -1.416666667 + 1.416666667 = 0.000000000
0.000000000 + z = -1.227576655 + 1.416666667
z = -1.227576655 + 1.416666667

Combine like terms: -1.227576655 + 1.416666667 = 0.189090012
z = 0.189090012

Simplifying
z = 0.189090012

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.644243322, 0.189090012}