(3z-2)(2z)=33

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Solution for (3z-2)(2z)=33 equation:


Simplifying
(3z + -2)(2z) = 33

Reorder the terms:
(-2 + 3z)(2z) = 33

Remove parenthesis around (2z)
(-2 + 3z) * 2z = 33

Reorder the terms for easier multiplication:
2z(-2 + 3z) = 33
(-2 * 2z + 3z * 2z) = 33
(-4z + 6z2) = 33

Solving
-4z + 6z2 = 33

Solving for variable 'z'.

Reorder the terms:
-33 + -4z + 6z2 = 33 + -33

Combine like terms: 33 + -33 = 0
-33 + -4z + 6z2 = 0

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

Divide each side by '6'.
-5.5 + -0.6666666667z + z2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + -0.6666666667z + 5.5 + z2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + -0.6666666667z + z2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + -0.6666666667z + z2 = 0 + 5.5
-0.6666666667z + z2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
-0.6666666667z + z2 = 5.5

The z term is -0.6666666667z.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667z + 0.1111111112 + z2 = 5.5 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667z + z2 = 5.5 + 0.1111111112

Combine like terms: 5.5 + 0.1111111112 = 5.6111111112
0.1111111112 + -0.6666666667z + z2 = 5.6111111112

Factor a perfect square on the left side:
(z + -0.3333333334)(z + -0.3333333334) = 5.6111111112

Calculate the square root of the right side: 2.368778401

Break this problem into two subproblems by setting 
(z + -0.3333333334) equal to 2.368778401 and -2.368778401.

Subproblem 1

z + -0.3333333334 = 2.368778401 Simplifying z + -0.3333333334 = 2.368778401 Reorder the terms: -0.3333333334 + z = 2.368778401 Solving -0.3333333334 + z = 2.368778401 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + z = 2.368778401 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + z = 2.368778401 + 0.3333333334 z = 2.368778401 + 0.3333333334 Combine like terms: 2.368778401 + 0.3333333334 = 2.7021117344 z = 2.7021117344 Simplifying z = 2.7021117344

Subproblem 2

z + -0.3333333334 = -2.368778401 Simplifying z + -0.3333333334 = -2.368778401 Reorder the terms: -0.3333333334 + z = -2.368778401 Solving -0.3333333334 + z = -2.368778401 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + z = -2.368778401 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + z = -2.368778401 + 0.3333333334 z = -2.368778401 + 0.3333333334 Combine like terms: -2.368778401 + 0.3333333334 = -2.0354450676 z = -2.0354450676 Simplifying z = -2.0354450676

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.7021117344, -2.0354450676}

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