z^2-z-9=0

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


Simplifying
z2 + -1z + -9 = 0

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

Solving
-9 + -1z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The z term is -1z.  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.
-1z + 0.25 + z2 = 9 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 9 + 0.25

Combine like terms: 9 + 0.25 = 9.25
0.25 + -1z + z2 = 9.25

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(z + -0.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

z + -0.5 = 3.041381265 Simplifying z + -0.5 = 3.041381265 Reorder the terms: -0.5 + z = 3.041381265 Solving -0.5 + z = 3.041381265 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 = 3.041381265 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 3.041381265 + 0.5 z = 3.041381265 + 0.5 Combine like terms: 3.041381265 + 0.5 = 3.541381265 z = 3.541381265 Simplifying z = 3.541381265

Subproblem 2

z + -0.5 = -3.041381265 Simplifying z + -0.5 = -3.041381265 Reorder the terms: -0.5 + z = -3.041381265 Solving -0.5 + z = -3.041381265 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 = -3.041381265 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -3.041381265 + 0.5 z = -3.041381265 + 0.5 Combine like terms: -3.041381265 + 0.5 = -2.541381265 z = -2.541381265 Simplifying z = -2.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.541381265, -2.541381265}

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