8z(14-3z)=8

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Solution for 8z(14-3z)=8 equation: 


Simplifying
8z(14 + -3z) = 8
(14 * 8z + -3z * 8z) = 8
(112z + -24z2) = 8

Solving
112z + -24z2 = 8

Solving for variable 'z'.

Reorder the terms:
-8 + 112z + -24z2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 112z + -24z2 = 0

Factor out the Greatest Common Factor (GCF), '8'.
8(-1 + 14z + -3z2) = 0

Ignore the factor 8.

Subproblem 1
Set the factor '(-1 + 14z + -3z2)' equal to zero and attempt to solve:

Simplifying
-1 + 14z + -3z2 = 0

Solving
-1 + 14z + -3z2 = 0

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

Divide each side by '-3'.
0.3333333333 + -4.666666667z + z2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -4.666666667z + -0.3333333333 + z2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -4.666666667z + z2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -4.666666667z + z2 = 0 + -0.3333333333
-4.666666667z + z2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-4.666666667z + z2 = -0.3333333333

The z term is -4.666666667z.  Take half its coefficient (-2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
-4.666666667z + 5.444444448 + z2 = -0.3333333333 + 5.444444448

Reorder the terms:
5.444444448 + -4.666666667z + z2 = -0.3333333333 + 5.444444448

Combine like terms: -0.3333333333 + 5.444444448 = 5.1111111147
5.444444448 + -4.666666667z + z2 = 5.1111111147

Factor a perfect square on the left side:
(z + -2.333333334)(z + -2.333333334) = 5.1111111147

Calculate the square root of the right side: 2.260776662

Break this problem into two subproblems by setting 
(z + -2.333333334) equal to 2.260776662 and -2.260776662.


Subproblem 1
z + -2.333333334 = 2.260776662

Simplifying
z + -2.333333334 = 2.260776662

Reorder the terms:
-2.333333334 + z = 2.260776662

Solving
-2.333333334 + z = 2.260776662

Solving for variable 'z'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + z = 2.260776662 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + z = 2.260776662 + 2.333333334
z = 2.260776662 + 2.333333334

Combine like terms: 2.260776662 + 2.333333334 = 4.594109996
z = 4.594109996

Simplifying
z = 4.594109996


Subproblem 2
z + -2.333333334 = -2.260776662

Simplifying
z + -2.333333334 = -2.260776662

Reorder the terms:
-2.333333334 + z = -2.260776662

Solving
-2.333333334 + z = -2.260776662

Solving for variable 'z'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + z = -2.260776662 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + z = -2.260776662 + 2.333333334
z = -2.260776662 + 2.333333334

Combine like terms: -2.260776662 + 2.333333334 = 0.072556672
z = 0.072556672

Simplifying
z = 0.072556672


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.594109996, 0.072556672}
Solution
z = {4.594109996, 0.072556672}

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