3z^2-7z=3

Solution for 3z^2-7z=3 equation:

Simplifying
3z2 + -7z = 3

Reorder the terms:
-7z + 3z2 = 3

Solving
-7z + 3z2 = 3

Solving for variable 'z'.

Reorder the terms:
-3 + -7z + 3z2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + -7z + 3z2 = 0

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

Divide each side by '3'.
-1 + -2.333333333z + z2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2.333333333z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + -2.333333333z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2.333333333z + z2 = 0 + 1
-2.333333333z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2.333333333z + z2 = 1

The z term is -2.333333333z.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333z + 1.361111112 + z2 = 1 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333z + z2 = 1 + 1.361111112

Combine like terms: 1 + 1.361111112 = 2.361111112
1.361111112 + -2.333333333z + z2 = 2.361111112

Factor a perfect square on the left side:
(z + -1.166666667)(z + -1.166666667) = 2.361111112

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(z + -1.166666667) equal to 1.536590743 and -1.536590743.

Subproblem 1
z + -1.166666667 = 1.536590743

Simplifying
z + -1.166666667 = 1.536590743

Reorder the terms:
-1.166666667 + z = 1.536590743

Solving
-1.166666667 + z = 1.536590743

Solving for variable 'z'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + z = 1.536590743 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + z = 1.536590743 + 1.166666667
z = 1.536590743 + 1.166666667

Combine like terms: 1.536590743 + 1.166666667 = 2.70325741
z = 2.70325741

Simplifying
z = 2.70325741

Subproblem 2
z + -1.166666667 = -1.536590743

Simplifying
z + -1.166666667 = -1.536590743

Reorder the terms:
-1.166666667 + z = -1.536590743

Solving
-1.166666667 + z = -1.536590743

Solving for variable 'z'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + z = -1.536590743 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + z = -1.536590743 + 1.166666667
z = -1.536590743 + 1.166666667

Combine like terms: -1.536590743 + 1.166666667 = -0.369924076
z = -0.369924076

Simplifying
z = -0.369924076

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.70325741, -0.369924076}