9z(z-1)-2(3z+4)=7

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Solution for 9z(z-1)-2(3z+4)=7 equation: 


Simplifying
9z(z + -1) + -2(3z + 4) = 7

Reorder the terms:
9z(-1 + z) + -2(3z + 4) = 7
(-1 * 9z + z * 9z) + -2(3z + 4) = 7
(-9z + 9z2) + -2(3z + 4) = 7

Reorder the terms:
-9z + 9z2 + -2(4 + 3z) = 7
-9z + 9z2 + (4 * -2 + 3z * -2) = 7
-9z + 9z2 + (-8 + -6z) = 7

Reorder the terms:
-8 + -9z + -6z + 9z2 = 7

Combine like terms: -9z + -6z = -15z
-8 + -15z + 9z2 = 7

Solving
-8 + -15z + 9z2 = 7

Solving for variable 'z'.

Reorder the terms:
-8 + -7 + -15z + 9z2 = 7 + -7

Combine like terms: -8 + -7 = -15
-15 + -15z + 9z2 = 7 + -7

Combine like terms: 7 + -7 = 0
-15 + -15z + 9z2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-5 + -5z + 3z2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-5 + -5z + 3z2)' equal to zero and attempt to solve:

Simplifying
-5 + -5z + 3z2 = 0

Solving
-5 + -5z + 3z2 = 0

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

Divide each side by '3'.
-1.666666667 + -1.666666667z + z2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -1.666666667z + 1.666666667 + z2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -1.666666667z + z2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -1.666666667z + z2 = 0 + 1.666666667
-1.666666667z + z2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-1.666666667z + z2 = 1.666666667

The z term is -1.666666667z.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667z + 0.6944444447 + z2 = 1.666666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667z + z2 = 1.666666667 + 0.6944444447

Combine like terms: 1.666666667 + 0.6944444447 = 2.3611111117
0.6944444447 + -1.666666667z + z2 = 2.3611111117

Factor a perfect square on the left side:
(z + -0.8333333335)(z + -0.8333333335) = 2.3611111117

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(z + -0.8333333335) equal to 1.536590743 and -1.536590743.


Subproblem 1
z + -0.8333333335 = 1.536590743

Simplifying
z + -0.8333333335 = 1.536590743

Reorder the terms:
-0.8333333335 + z = 1.536590743

Solving
-0.8333333335 + z = 1.536590743

Solving for variable 'z'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + z = 1.536590743 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + z = 1.536590743 + 0.8333333335
z = 1.536590743 + 0.8333333335

Combine like terms: 1.536590743 + 0.8333333335 = 2.3699240765
z = 2.3699240765

Simplifying
z = 2.3699240765


Subproblem 2
z + -0.8333333335 = -1.536590743

Simplifying
z + -0.8333333335 = -1.536590743

Reorder the terms:
-0.8333333335 + z = -1.536590743

Solving
-0.8333333335 + z = -1.536590743

Solving for variable 'z'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + z = -1.536590743 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + z = -1.536590743 + 0.8333333335
z = -1.536590743 + 0.8333333335

Combine like terms: -1.536590743 + 0.8333333335 = -0.7032574095
z = -0.7032574095

Simplifying
z = -0.7032574095


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.3699240765, -0.7032574095}
Solution
z = {2.3699240765, -0.7032574095}

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