z(2z+6)=4

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


Simplifying
z(2z + 6) = 4

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

Solving
6z + 2z2 = 4

Solving for variable 'z'.

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

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

Factor out the Greatest Common Factor (GCF), '2'.
2(-2 + 3z + z2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-2 + 3z + z2)' equal to zero and attempt to solve:

Simplifying
-2 + 3z + z2 = 0

Solving
-2 + 3z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 3z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + 3z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 3z + z2 = 0 + 2
3z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
3z + z2 = 2

The z term is 3z.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3z + 2.25 + z2 = 2 + 2.25

Reorder the terms:
2.25 + 3z + z2 = 2 + 2.25

Combine like terms: 2 + 2.25 = 4.25
2.25 + 3z + z2 = 4.25

Factor a perfect square on the left side:
(z + 1.5)(z + 1.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(z + 1.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
z + 1.5 = 2.061552813

Simplifying
z + 1.5 = 2.061552813

Reorder the terms:
1.5 + z = 2.061552813

Solving
1.5 + z = 2.061552813

Solving for variable 'z'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + z = 2.061552813 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = 2.061552813 + -1.5
z = 2.061552813 + -1.5

Combine like terms: 2.061552813 + -1.5 = 0.561552813
z = 0.561552813

Simplifying
z = 0.561552813


Subproblem 2
z + 1.5 = -2.061552813

Simplifying
z + 1.5 = -2.061552813

Reorder the terms:
1.5 + z = -2.061552813

Solving
1.5 + z = -2.061552813

Solving for variable 'z'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + z = -2.061552813 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = -2.061552813 + -1.5
z = -2.061552813 + -1.5

Combine like terms: -2.061552813 + -1.5 = -3.561552813
z = -3.561552813

Simplifying
z = -3.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.561552813, -3.561552813}
Solution
z = {0.561552813, -3.561552813}

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