5=z(3z-1)

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Solution for 5=z(3z-1) equation:


Simplifying
5 = z(3z + -1)

Reorder the terms:
5 = z(-1 + 3z)
5 = (-1 * z + 3z * z)
5 = (-1z + 3z2)

Solving
5 = -1z + 3z2

Solving for variable 'z'.

Reorder the terms:
5 + z + -3z2 = -1z + z + 3z2 + -3z2

Combine like terms: -1z + z = 0
5 + z + -3z2 = 0 + 3z2 + -3z2
5 + z + -3z2 = 3z2 + -3z2

Combine like terms: 3z2 + -3z2 = 0
5 + z + -3z2 = 0

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

Divide each side by '-3'.
-1.666666667 + -0.3333333333z + z2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -0.3333333333z + 1.666666667 + z2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -0.3333333333z + z2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -0.3333333333z + z2 = 0 + 1.666666667
-0.3333333333z + z2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-0.3333333333z + z2 = 1.666666667

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

Add '0.25' to each side of the equation.
-0.3333333333z + 0.25 + z2 = 1.666666667 + 0.25

Reorder the terms:
0.25 + -0.3333333333z + z2 = 1.666666667 + 0.25

Combine like terms: 1.666666667 + 0.25 = 1.916666667
0.25 + -0.3333333333z + z2 = 1.916666667

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 1.916666667

Calculate the square root of the right side: 1.384437311

Break this problem into two subproblems by setting 
(z + 0.5) equal to 1.384437311 and -1.384437311.

Subproblem 1

z + 0.5 = 1.384437311 Simplifying z + 0.5 = 1.384437311 Reorder the terms: 0.5 + z = 1.384437311 Solving 0.5 + z = 1.384437311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 1.384437311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 1.384437311 + -0.5 z = 1.384437311 + -0.5 Combine like terms: 1.384437311 + -0.5 = 0.884437311 z = 0.884437311 Simplifying z = 0.884437311

Subproblem 2

z + 0.5 = -1.384437311 Simplifying z + 0.5 = -1.384437311 Reorder the terms: 0.5 + z = -1.384437311 Solving 0.5 + z = -1.384437311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -1.384437311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -1.384437311 + -0.5 z = -1.384437311 + -0.5 Combine like terms: -1.384437311 + -0.5 = -1.884437311 z = -1.884437311 Simplifying z = -1.884437311

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.884437311, -1.884437311}

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