(6z+5)(z-8)=8

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Solution for (6z+5)(z-8)=8 equation:


Simplifying
(6z + 5)(z + -8) = 8

Reorder the terms:
(5 + 6z)(z + -8) = 8

Reorder the terms:
(5 + 6z)(-8 + z) = 8

Multiply (5 + 6z) * (-8 + z)
(5(-8 + z) + 6z * (-8 + z)) = 8
((-8 * 5 + z * 5) + 6z * (-8 + z)) = 8
((-40 + 5z) + 6z * (-8 + z)) = 8
(-40 + 5z + (-8 * 6z + z * 6z)) = 8
(-40 + 5z + (-48z + 6z2)) = 8

Combine like terms: 5z + -48z = -43z
(-40 + -43z + 6z2) = 8

Solving
-40 + -43z + 6z2 = 8

Solving for variable 'z'.

Reorder the terms:
-40 + -8 + -43z + 6z2 = 8 + -8

Combine like terms: -40 + -8 = -48
-48 + -43z + 6z2 = 8 + -8

Combine like terms: 8 + -8 = 0
-48 + -43z + 6z2 = 0

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

Divide each side by '6'.
-8 + -7.166666667z + z2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -7.166666667z + 8 + z2 = 0 + 8

Reorder the terms:
-8 + 8 + -7.166666667z + z2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -7.166666667z + z2 = 0 + 8
-7.166666667z + z2 = 0 + 8

Combine like terms: 0 + 8 = 8
-7.166666667z + z2 = 8

The z term is -7.166666667z.  Take half its coefficient (-3.583333334).
Square it (12.84027778) and add it to both sides.

Add '12.84027778' to each side of the equation.
-7.166666667z + 12.84027778 + z2 = 8 + 12.84027778

Reorder the terms:
12.84027778 + -7.166666667z + z2 = 8 + 12.84027778

Combine like terms: 8 + 12.84027778 = 20.84027778
12.84027778 + -7.166666667z + z2 = 20.84027778

Factor a perfect square on the left side:
(z + -3.583333334)(z + -3.583333334) = 20.84027778

Calculate the square root of the right side: 4.565115309

Break this problem into two subproblems by setting 
(z + -3.583333334) equal to 4.565115309 and -4.565115309.

Subproblem 1

z + -3.583333334 = 4.565115309 Simplifying z + -3.583333334 = 4.565115309 Reorder the terms: -3.583333334 + z = 4.565115309 Solving -3.583333334 + z = 4.565115309 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.583333334' to each side of the equation. -3.583333334 + 3.583333334 + z = 4.565115309 + 3.583333334 Combine like terms: -3.583333334 + 3.583333334 = 0.000000000 0.000000000 + z = 4.565115309 + 3.583333334 z = 4.565115309 + 3.583333334 Combine like terms: 4.565115309 + 3.583333334 = 8.148448643 z = 8.148448643 Simplifying z = 8.148448643

Subproblem 2

z + -3.583333334 = -4.565115309 Simplifying z + -3.583333334 = -4.565115309 Reorder the terms: -3.583333334 + z = -4.565115309 Solving -3.583333334 + z = -4.565115309 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.583333334' to each side of the equation. -3.583333334 + 3.583333334 + z = -4.565115309 + 3.583333334 Combine like terms: -3.583333334 + 3.583333334 = 0.000000000 0.000000000 + z = -4.565115309 + 3.583333334 z = -4.565115309 + 3.583333334 Combine like terms: -4.565115309 + 3.583333334 = -0.981781975 z = -0.981781975 Simplifying z = -0.981781975

Solution

The solution to the problem is based on the solutions from the subproblems. z = {8.148448643, -0.981781975}

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