z^2-3z-9=0

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Solution for z^2-3z-9=0 equation:


Simplifying
z2 + -3z + -9 = 0

Reorder the terms:
-9 + -3z + z2 = 0

Solving
-9 + -3z + 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 + -3z + 9 + z2 = 0 + 9

Reorder the terms:
-9 + 9 + -3z + z2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + -3z + z2 = 0 + 9
-3z + z2 = 0 + 9

Combine like terms: 0 + 9 = 9
-3z + z2 = 9

The z term is -3z.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3z + 2.25 + z2 = 9 + 2.25

Reorder the terms:
2.25 + -3z + z2 = 9 + 2.25

Combine like terms: 9 + 2.25 = 11.25
2.25 + -3z + z2 = 11.25

Factor a perfect square on the left side:
(z + -1.5)(z + -1.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(z + -1.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

z + -1.5 = 3.354101966 Simplifying z + -1.5 = 3.354101966 Reorder the terms: -1.5 + z = 3.354101966 Solving -1.5 + z = 3.354101966 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = 3.354101966 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = 3.354101966 + 1.5 z = 3.354101966 + 1.5 Combine like terms: 3.354101966 + 1.5 = 4.854101966 z = 4.854101966 Simplifying z = 4.854101966

Subproblem 2

z + -1.5 = -3.354101966 Simplifying z + -1.5 = -3.354101966 Reorder the terms: -1.5 + z = -3.354101966 Solving -1.5 + z = -3.354101966 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = -3.354101966 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = -3.354101966 + 1.5 z = -3.354101966 + 1.5 Combine like terms: -3.354101966 + 1.5 = -1.854101966 z = -1.854101966 Simplifying z = -1.854101966

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.854101966, -1.854101966}

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