(3z-2)(8+z)=2

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Solution for (3z-2)(8+z)=2 equation:


Simplifying
(3z + -2)(8 + z) = 2

Reorder the terms:
(-2 + 3z)(8 + z) = 2

Multiply (-2 + 3z) * (8 + z)
(-2(8 + z) + 3z * (8 + z)) = 2
((8 * -2 + z * -2) + 3z * (8 + z)) = 2
((-16 + -2z) + 3z * (8 + z)) = 2
(-16 + -2z + (8 * 3z + z * 3z)) = 2
(-16 + -2z + (24z + 3z2)) = 2

Combine like terms: -2z + 24z = 22z
(-16 + 22z + 3z2) = 2

Solving
-16 + 22z + 3z2 = 2

Solving for variable 'z'.

Reorder the terms:
-16 + -2 + 22z + 3z2 = 2 + -2

Combine like terms: -16 + -2 = -18
-18 + 22z + 3z2 = 2 + -2

Combine like terms: 2 + -2 = 0
-18 + 22z + 3z2 = 0

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

Divide each side by '3'.
-6 + 7.333333333z + z2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 7.333333333z + 6 + z2 = 0 + 6

Reorder the terms:
-6 + 6 + 7.333333333z + z2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 7.333333333z + z2 = 0 + 6
7.333333333z + z2 = 0 + 6

Combine like terms: 0 + 6 = 6
7.333333333z + z2 = 6

The z term is 7.333333333z.  Take half its coefficient (3.666666667).
Square it (13.44444445) and add it to both sides.

Add '13.44444445' to each side of the equation.
7.333333333z + 13.44444445 + z2 = 6 + 13.44444445

Reorder the terms:
13.44444445 + 7.333333333z + z2 = 6 + 13.44444445

Combine like terms: 6 + 13.44444445 = 19.44444445
13.44444445 + 7.333333333z + z2 = 19.44444445

Factor a perfect square on the left side:
(z + 3.666666667)(z + 3.666666667) = 19.44444445

Calculate the square root of the right side: 4.409585519

Break this problem into two subproblems by setting 
(z + 3.666666667) equal to 4.409585519 and -4.409585519.

Subproblem 1

z + 3.666666667 = 4.409585519 Simplifying z + 3.666666667 = 4.409585519 Reorder the terms: 3.666666667 + z = 4.409585519 Solving 3.666666667 + z = 4.409585519 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.666666667' to each side of the equation. 3.666666667 + -3.666666667 + z = 4.409585519 + -3.666666667 Combine like terms: 3.666666667 + -3.666666667 = 0.000000000 0.000000000 + z = 4.409585519 + -3.666666667 z = 4.409585519 + -3.666666667 Combine like terms: 4.409585519 + -3.666666667 = 0.742918852 z = 0.742918852 Simplifying z = 0.742918852

Subproblem 2

z + 3.666666667 = -4.409585519 Simplifying z + 3.666666667 = -4.409585519 Reorder the terms: 3.666666667 + z = -4.409585519 Solving 3.666666667 + z = -4.409585519 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.666666667' to each side of the equation. 3.666666667 + -3.666666667 + z = -4.409585519 + -3.666666667 Combine like terms: 3.666666667 + -3.666666667 = 0.000000000 0.000000000 + z = -4.409585519 + -3.666666667 z = -4.409585519 + -3.666666667 Combine like terms: -4.409585519 + -3.666666667 = -8.076252186 z = -8.076252186 Simplifying z = -8.076252186

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.742918852, -8.076252186}

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