2z(z-3)=6z+4

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Solution for 2z(z-3)=6z+4 equation: 


Simplifying
2z(z + -3) = 6z + 4

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

Reorder the terms:
-6z + 2z2 = 4 + 6z

Solving
-6z + 2z2 = 4 + 6z

Solving for variable 'z'.

Reorder the terms:
-4 + -6z + -6z + 2z2 = 4 + 6z + -4 + -6z

Combine like terms: -6z + -6z = -12z
-4 + -12z + 2z2 = 4 + 6z + -4 + -6z

Reorder the terms:
-4 + -12z + 2z2 = 4 + -4 + 6z + -6z

Combine like terms: 4 + -4 = 0
-4 + -12z + 2z2 = 0 + 6z + -6z
-4 + -12z + 2z2 = 6z + -6z

Combine like terms: 6z + -6z = 0
-4 + -12z + 2z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-2 + -6z + z2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-2 + -6z + z2)' equal to zero and attempt to solve:

Simplifying
-2 + -6z + z2 = 0

Solving
-2 + -6z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -6z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + -6z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -6z + z2 = 0 + 2
-6z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
-6z + z2 = 2

The z term is -6z.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6z + 9 + z2 = 2 + 9

Reorder the terms:
9 + -6z + z2 = 2 + 9

Combine like terms: 2 + 9 = 11
9 + -6z + z2 = 11

Factor a perfect square on the left side:
(z + -3)(z + -3) = 11

Calculate the square root of the right side: 3.31662479

Break this problem into two subproblems by setting 
(z + -3) equal to 3.31662479 and -3.31662479.


Subproblem 1
z + -3 = 3.31662479

Simplifying
z + -3 = 3.31662479

Reorder the terms:
-3 + z = 3.31662479

Solving
-3 + z = 3.31662479

Solving for variable 'z'.

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

Add '3' to each side of the equation.
-3 + 3 + z = 3.31662479 + 3

Combine like terms: -3 + 3 = 0
0 + z = 3.31662479 + 3
z = 3.31662479 + 3

Combine like terms: 3.31662479 + 3 = 6.31662479
z = 6.31662479

Simplifying
z = 6.31662479


Subproblem 2
z + -3 = -3.31662479

Simplifying
z + -3 = -3.31662479

Reorder the terms:
-3 + z = -3.31662479

Solving
-3 + z = -3.31662479

Solving for variable 'z'.

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

Add '3' to each side of the equation.
-3 + 3 + z = -3.31662479 + 3

Combine like terms: -3 + 3 = 0
0 + z = -3.31662479 + 3
z = -3.31662479 + 3

Combine like terms: -3.31662479 + 3 = -0.31662479
z = -0.31662479

Simplifying
z = -0.31662479


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {6.31662479, -0.31662479}
Solution
z = {6.31662479, -0.31662479}

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