8z^2-9z-82=0

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Solution for 8z^2-9z-82=0 equation:


Simplifying
8z2 + -9z + -82 = 0

Reorder the terms:
-82 + -9z + 8z2 = 0

Solving
-82 + -9z + 8z2 = 0

Solving for variable 'z'.

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

Divide each side by '8'.
-10.25 + -1.125z + z2 = 0

Move the constant term to the right:

Add '10.25' to each side of the equation.
-10.25 + -1.125z + 10.25 + z2 = 0 + 10.25

Reorder the terms:
-10.25 + 10.25 + -1.125z + z2 = 0 + 10.25

Combine like terms: -10.25 + 10.25 = 0.00
0.00 + -1.125z + z2 = 0 + 10.25
-1.125z + z2 = 0 + 10.25

Combine like terms: 0 + 10.25 = 10.25
-1.125z + z2 = 10.25

The z term is -1.125z.  Take half its coefficient (-0.5625).
Square it (0.31640625) and add it to both sides.

Add '0.31640625' to each side of the equation.
-1.125z + 0.31640625 + z2 = 10.25 + 0.31640625

Reorder the terms:
0.31640625 + -1.125z + z2 = 10.25 + 0.31640625

Combine like terms: 10.25 + 0.31640625 = 10.56640625
0.31640625 + -1.125z + z2 = 10.56640625

Factor a perfect square on the left side:
(z + -0.5625)(z + -0.5625) = 10.56640625

Calculate the square root of the right side: 3.250600906

Break this problem into two subproblems by setting 
(z + -0.5625) equal to 3.250600906 and -3.250600906.

Subproblem 1

z + -0.5625 = 3.250600906 Simplifying z + -0.5625 = 3.250600906 Reorder the terms: -0.5625 + z = 3.250600906 Solving -0.5625 + z = 3.250600906 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + z = 3.250600906 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + z = 3.250600906 + 0.5625 z = 3.250600906 + 0.5625 Combine like terms: 3.250600906 + 0.5625 = 3.813100906 z = 3.813100906 Simplifying z = 3.813100906

Subproblem 2

z + -0.5625 = -3.250600906 Simplifying z + -0.5625 = -3.250600906 Reorder the terms: -0.5625 + z = -3.250600906 Solving -0.5625 + z = -3.250600906 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + z = -3.250600906 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + z = -3.250600906 + 0.5625 z = -3.250600906 + 0.5625 Combine like terms: -3.250600906 + 0.5625 = -2.688100906 z = -2.688100906 Simplifying z = -2.688100906

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.813100906, -2.688100906}

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