3z-6z^2=-8

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Solution for 3z-6z^2=-8 equation:


Simplifying
3z + -6z2 = -8

Solving
3z + -6z2 = -8

Solving for variable 'z'.

Reorder the terms:
8 + 3z + -6z2 = -8 + 8

Combine like terms: -8 + 8 = 0
8 + 3z + -6z2 = 0

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

Divide each side by '-6'.
-1.333333333 + -0.5z + z2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -0.5z + 1.333333333 + z2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -0.5z + z2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -0.5z + z2 = 0 + 1.333333333
-0.5z + z2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-0.5z + z2 = 1.333333333

The z term is -0.5z.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5z + 0.0625 + z2 = 1.333333333 + 0.0625

Reorder the terms:
0.0625 + -0.5z + z2 = 1.333333333 + 0.0625

Combine like terms: 1.333333333 + 0.0625 = 1.395833333
0.0625 + -0.5z + z2 = 1.395833333

Factor a perfect square on the left side:
(z + -0.25)(z + -0.25) = 1.395833333

Calculate the square root of the right side: 1.181453906

Break this problem into two subproblems by setting 
(z + -0.25) equal to 1.181453906 and -1.181453906.

Subproblem 1

z + -0.25 = 1.181453906 Simplifying z + -0.25 = 1.181453906 Reorder the terms: -0.25 + z = 1.181453906 Solving -0.25 + z = 1.181453906 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = 1.181453906 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = 1.181453906 + 0.25 z = 1.181453906 + 0.25 Combine like terms: 1.181453906 + 0.25 = 1.431453906 z = 1.431453906 Simplifying z = 1.431453906

Subproblem 2

z + -0.25 = -1.181453906 Simplifying z + -0.25 = -1.181453906 Reorder the terms: -0.25 + z = -1.181453906 Solving -0.25 + z = -1.181453906 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = -1.181453906 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = -1.181453906 + 0.25 z = -1.181453906 + 0.25 Combine like terms: -1.181453906 + 0.25 = -0.931453906 z = -0.931453906 Simplifying z = -0.931453906

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.431453906, -0.931453906}

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