22z+22=14z^2+30z-48

Solution for 22z+22=14z^2+30z-48 equation:

Simplifying
22z + 22 = 14z2 + 30z + -48

Reorder the terms:
22 + 22z = 14z2 + 30z + -48

Reorder the terms:
22 + 22z = -48 + 30z + 14z2

Solving
22 + 22z = -48 + 30z + 14z2

Solving for variable 'z'.

Reorder the terms:
22 + 48 + 22z + -30z + -14z2 = -48 + 30z + 14z2 + 48 + -30z + -14z2

Combine like terms: 22 + 48 = 70
70 + 22z + -30z + -14z2 = -48 + 30z + 14z2 + 48 + -30z + -14z2

Combine like terms: 22z + -30z = -8z
70 + -8z + -14z2 = -48 + 30z + 14z2 + 48 + -30z + -14z2

Reorder the terms:
70 + -8z + -14z2 = -48 + 48 + 30z + -30z + 14z2 + -14z2

Combine like terms: -48 + 48 = 0
70 + -8z + -14z2 = 0 + 30z + -30z + 14z2 + -14z2
70 + -8z + -14z2 = 30z + -30z + 14z2 + -14z2

Combine like terms: 30z + -30z = 0
70 + -8z + -14z2 = 0 + 14z2 + -14z2
70 + -8z + -14z2 = 14z2 + -14z2

Combine like terms: 14z2 + -14z2 = 0
70 + -8z + -14z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(35 + -4z + -7z2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(35 + -4z + -7z2)' equal to zero and attempt to solve:

Simplifying
35 + -4z + -7z2 = 0

Solving
35 + -4z + -7z2 = 0

Begin completing the square.  Divide all terms by
-7 the coefficient of the squared term: 

Divide each side by '-7'.
-5 + 0.5714285714z + z2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 0.5714285714z + 5 + z2 = 0 + 5

Reorder the terms:
-5 + 5 + 0.5714285714z + z2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 0.5714285714z + z2 = 0 + 5
0.5714285714z + z2 = 0 + 5

Combine like terms: 0 + 5 = 5
0.5714285714z + z2 = 5

The z term is 0.5714285714z.  Take half its coefficient (0.2857142857).
Square it (0.08163265305) and add it to both sides.

Add '0.08163265305' to each side of the equation.
0.5714285714z + 0.08163265305 + z2 = 5 + 0.08163265305

Reorder the terms:
0.08163265305 + 0.5714285714z + z2 = 5 + 0.08163265305

Combine like terms: 5 + 0.08163265305 = 5.08163265305
0.08163265305 + 0.5714285714z + z2 = 5.08163265305

Factor a perfect square on the left side:
(z + 0.2857142857)(z + 0.2857142857) = 5.08163265305

Calculate the square root of the right side: 2.254247691

Break this problem into two subproblems by setting 
(z + 0.2857142857) equal to 2.254247691 and -2.254247691.

Subproblem 1
z + 0.2857142857 = 2.254247691

Simplifying
z + 0.2857142857 = 2.254247691

Reorder the terms:
0.2857142857 + z = 2.254247691

Solving
0.2857142857 + z = 2.254247691

Solving for variable 'z'.

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

Add '-0.2857142857' to each side of the equation.
0.2857142857 + -0.2857142857 + z = 2.254247691 + -0.2857142857

Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000
0.0000000000 + z = 2.254247691 + -0.2857142857
z = 2.254247691 + -0.2857142857

Combine like terms: 2.254247691 + -0.2857142857 = 1.9685334053
z = 1.9685334053

Simplifying
z = 1.9685334053

Subproblem 2
z + 0.2857142857 = -2.254247691

Simplifying
z + 0.2857142857 = -2.254247691

Reorder the terms:
0.2857142857 + z = -2.254247691

Solving
0.2857142857 + z = -2.254247691

Solving for variable 'z'.

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

Add '-0.2857142857' to each side of the equation.
0.2857142857 + -0.2857142857 + z = -2.254247691 + -0.2857142857

Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000
0.0000000000 + z = -2.254247691 + -0.2857142857
z = -2.254247691 + -0.2857142857

Combine like terms: -2.254247691 + -0.2857142857 = -2.5399619767
z = -2.5399619767

Simplifying
z = -2.5399619767

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1.9685334053, -2.5399619767}
Solution
z = {1.9685334053, -2.5399619767}