7z^2-3z=10-(9z^2+9z-4)

Solution for 7z^2-3z=10-(9z^2+9z-4) equation:

Simplifying
7z2 + -3z = 10 + -1(9z2 + 9z + -4)

Reorder the terms:
-3z + 7z2 = 10 + -1(9z2 + 9z + -4)

Reorder the terms:
-3z + 7z2 = 10 + -1(-4 + 9z + 9z2)
-3z + 7z2 = 10 + (-4 * -1 + 9z * -1 + 9z2 * -1)
-3z + 7z2 = 10 + (4 + -9z + -9z2)

Combine like terms: 10 + 4 = 14
-3z + 7z2 = 14 + -9z + -9z2

Solving
-3z + 7z2 = 14 + -9z + -9z2

Solving for variable 'z'.

Reorder the terms:
-14 + -3z + 9z + 7z2 + 9z2 = 14 + -9z + -9z2 + -14 + 9z + 9z2

Combine like terms: -3z + 9z = 6z
-14 + 6z + 7z2 + 9z2 = 14 + -9z + -9z2 + -14 + 9z + 9z2

Combine like terms: 7z2 + 9z2 = 16z2
-14 + 6z + 16z2 = 14 + -9z + -9z2 + -14 + 9z + 9z2

Reorder the terms:
-14 + 6z + 16z2 = 14 + -14 + -9z + 9z + -9z2 + 9z2

Combine like terms: 14 + -14 = 0
-14 + 6z + 16z2 = 0 + -9z + 9z + -9z2 + 9z2
-14 + 6z + 16z2 = -9z + 9z + -9z2 + 9z2

Combine like terms: -9z + 9z = 0
-14 + 6z + 16z2 = 0 + -9z2 + 9z2
-14 + 6z + 16z2 = -9z2 + 9z2

Combine like terms: -9z2 + 9z2 = 0
-14 + 6z + 16z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-7 + 3z + 8z2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-7 + 3z + 8z2)' equal to zero and attempt to solve:

Simplifying
-7 + 3z + 8z2 = 0

Solving
-7 + 3z + 8z2 = 0

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

Divide each side by '8'.
-0.875 + 0.375z + z2 = 0

Move the constant term to the right:

Add '0.875' to each side of the equation.
-0.875 + 0.375z + 0.875 + z2 = 0 + 0.875

Reorder the terms:
-0.875 + 0.875 + 0.375z + z2 = 0 + 0.875

Combine like terms: -0.875 + 0.875 = 0.000
0.000 + 0.375z + z2 = 0 + 0.875
0.375z + z2 = 0 + 0.875

Combine like terms: 0 + 0.875 = 0.875
0.375z + z2 = 0.875

The z term is 0.375z.  Take half its coefficient (0.1875).
Square it (0.03515625) and add it to both sides.

Add '0.03515625' to each side of the equation.
0.375z + 0.03515625 + z2 = 0.875 + 0.03515625

Reorder the terms:
0.03515625 + 0.375z + z2 = 0.875 + 0.03515625

Combine like terms: 0.875 + 0.03515625 = 0.91015625
0.03515625 + 0.375z + z2 = 0.91015625

Factor a perfect square on the left side:
(z + 0.1875)(z + 0.1875) = 0.91015625

Calculate the square root of the right side: 0.954021095

Break this problem into two subproblems by setting 
(z + 0.1875) equal to 0.954021095 and -0.954021095.

Subproblem 1
z + 0.1875 = 0.954021095

Simplifying
z + 0.1875 = 0.954021095

Reorder the terms:
0.1875 + z = 0.954021095

Solving
0.1875 + z = 0.954021095

Solving for variable 'z'.

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

Add '-0.1875' to each side of the equation.
0.1875 + -0.1875 + z = 0.954021095 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + z = 0.954021095 + -0.1875
z = 0.954021095 + -0.1875

Combine like terms: 0.954021095 + -0.1875 = 0.766521095
z = 0.766521095

Simplifying
z = 0.766521095

Subproblem 2
z + 0.1875 = -0.954021095

Simplifying
z + 0.1875 = -0.954021095

Reorder the terms:
0.1875 + z = -0.954021095

Solving
0.1875 + z = -0.954021095

Solving for variable 'z'.

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

Add '-0.1875' to each side of the equation.
0.1875 + -0.1875 + z = -0.954021095 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + z = -0.954021095 + -0.1875
z = -0.954021095 + -0.1875

Combine like terms: -0.954021095 + -0.1875 = -1.141521095
z = -1.141521095

Simplifying
z = -1.141521095

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.766521095, -1.141521095}
Solution
z = {0.766521095, -1.141521095}