z^2=9z-1

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Solution for z^2=9z-1 equation:


Simplifying
z2 = 9z + -1

Reorder the terms:
z2 = -1 + 9z

Solving
z2 = -1 + 9z

Solving for variable 'z'.

Reorder the terms:
1 + -9z + z2 = -1 + 9z + 1 + -9z

Reorder the terms:
1 + -9z + z2 = -1 + 1 + 9z + -9z

Combine like terms: -1 + 1 = 0
1 + -9z + z2 = 0 + 9z + -9z
1 + -9z + z2 = 9z + -9z

Combine like terms: 9z + -9z = 0
1 + -9z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -9z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + -9z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -9z + z2 = 0 + -1
-9z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
-9z + z2 = -1

The z term is -9z.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9z + 20.25 + z2 = -1 + 20.25

Reorder the terms:
20.25 + -9z + z2 = -1 + 20.25

Combine like terms: -1 + 20.25 = 19.25
20.25 + -9z + z2 = 19.25

Factor a perfect square on the left side:
(z + -4.5)(z + -4.5) = 19.25

Calculate the square root of the right side: 4.387482194

Break this problem into two subproblems by setting 
(z + -4.5) equal to 4.387482194 and -4.387482194.

Subproblem 1

z + -4.5 = 4.387482194 Simplifying z + -4.5 = 4.387482194 Reorder the terms: -4.5 + z = 4.387482194 Solving -4.5 + z = 4.387482194 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + z = 4.387482194 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = 4.387482194 + 4.5 z = 4.387482194 + 4.5 Combine like terms: 4.387482194 + 4.5 = 8.887482194 z = 8.887482194 Simplifying z = 8.887482194

Subproblem 2

z + -4.5 = -4.387482194 Simplifying z + -4.5 = -4.387482194 Reorder the terms: -4.5 + z = -4.387482194 Solving -4.5 + z = -4.387482194 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + z = -4.387482194 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = -4.387482194 + 4.5 z = -4.387482194 + 4.5 Combine like terms: -4.387482194 + 4.5 = 0.112517806 z = 0.112517806 Simplifying z = 0.112517806

Solution

The solution to the problem is based on the solutions from the subproblems. z = {8.887482194, 0.112517806}

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