2z^2=2(z+3)

Solution for 2z^2=2(z+3) equation:

Simplifying
2z2 = 2(z + 3)

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

Solving
2z2 = 6 + 2z

Solving for variable 'z'.

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

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

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

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

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + -1z + z2) = 0

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

Simplifying
-3 + -1z + z2 = 0

Solving
-3 + -1z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -1z + 3 + z2 = 0 + 3

Reorder the terms:
-3 + 3 + -1z + z2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1z + z2 = 0 + 3
-1z + z2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1z + z2 = 3

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = 3 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 3 + 0.25

Combine like terms: 3 + 0.25 = 3.25
0.25 + -1z + z2 = 3.25

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(z + -0.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
z + -0.5 = 1.802775638

Simplifying
z + -0.5 = 1.802775638

Reorder the terms:
-0.5 + z = 1.802775638

Solving
-0.5 + z = 1.802775638

Solving for variable 'z'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = 1.802775638 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = 1.802775638 + 0.5
z = 1.802775638 + 0.5

Combine like terms: 1.802775638 + 0.5 = 2.302775638
z = 2.302775638

Simplifying
z = 2.302775638

Subproblem 2
z + -0.5 = -1.802775638

Simplifying
z + -0.5 = -1.802775638

Reorder the terms:
-0.5 + z = -1.802775638

Solving
-0.5 + z = -1.802775638

Solving for variable 'z'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = -1.802775638 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = -1.802775638 + 0.5
z = -1.802775638 + 0.5

Combine like terms: -1.802775638 + 0.5 = -1.302775638
z = -1.302775638

Simplifying
z = -1.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.302775638, -1.302775638}
Solution
z = {2.302775638, -1.302775638}