z^2=6z-2

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


Simplifying
z2 = 6z + -2

Reorder the terms:
z2 = -2 + 6z

Solving
z2 = -2 + 6z

Solving for variable 'z'.

Reorder the terms:
2 + -6z + z2 = -2 + 6z + 2 + -6z

Reorder the terms:
2 + -6z + z2 = -2 + 2 + 6z + -6z

Combine like terms: -2 + 2 = 0
2 + -6z + z2 = 0 + 6z + -6z
2 + -6z + z2 = 6z + -6z

Combine like terms: 6z + -6z = 0
2 + -6z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -6z + -2 + z2 = 0 + -2

Reorder the terms:
2 + -2 + -6z + z2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -6z + z2 = 0 + -2
-6z + z2 = 0 + -2

Combine like terms: 0 + -2 = -2
-6z + z2 = -2

The z term is -6z.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6z + 9 + z2 = -2 + 9

Reorder the terms:
9 + -6z + z2 = -2 + 9

Combine like terms: -2 + 9 = 7
9 + -6z + z2 = 7

Factor a perfect square on the left side:
(z + -3)(z + -3) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(z + -3) equal to 2.645751311 and -2.645751311.

Subproblem 1

z + -3 = 2.645751311 Simplifying z + -3 = 2.645751311 Reorder the terms: -3 + z = 2.645751311 Solving -3 + z = 2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 2.645751311 + 3 Combine like terms: -3 + 3 = 0 0 + z = 2.645751311 + 3 z = 2.645751311 + 3 Combine like terms: 2.645751311 + 3 = 5.645751311 z = 5.645751311 Simplifying z = 5.645751311

Subproblem 2

z + -3 = -2.645751311 Simplifying z + -3 = -2.645751311 Reorder the terms: -3 + z = -2.645751311 Solving -3 + z = -2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -2.645751311 + 3 Combine like terms: -3 + 3 = 0 0 + z = -2.645751311 + 3 z = -2.645751311 + 3 Combine like terms: -2.645751311 + 3 = 0.354248689 z = 0.354248689 Simplifying z = 0.354248689

Solution

The solution to the problem is based on the solutions from the subproblems. z = {5.645751311, 0.354248689}

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