z^2-9z+12=0

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Solution for z^2-9z+12=0 equation:


Simplifying
z2 + -9z + 12 = 0

Reorder the terms:
12 + -9z + z2 = 0

Solving
12 + -9z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + -9z + -12 + z2 = 0 + -12

Reorder the terms:
12 + -12 + -9z + z2 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + -9z + z2 = 0 + -12
-9z + z2 = 0 + -12

Combine like terms: 0 + -12 = -12
-9z + z2 = -12

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 = -12 + 20.25

Reorder the terms:
20.25 + -9z + z2 = -12 + 20.25

Combine like terms: -12 + 20.25 = 8.25
20.25 + -9z + z2 = 8.25

Factor a perfect square on the left side:
(z + -4.5)(z + -4.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(z + -4.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

z + -4.5 = 2.872281323 Simplifying z + -4.5 = 2.872281323 Reorder the terms: -4.5 + z = 2.872281323 Solving -4.5 + z = 2.872281323 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 = 2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = 2.872281323 + 4.5 z = 2.872281323 + 4.5 Combine like terms: 2.872281323 + 4.5 = 7.372281323 z = 7.372281323 Simplifying z = 7.372281323

Subproblem 2

z + -4.5 = -2.872281323 Simplifying z + -4.5 = -2.872281323 Reorder the terms: -4.5 + z = -2.872281323 Solving -4.5 + z = -2.872281323 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 = -2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = -2.872281323 + 4.5 z = -2.872281323 + 4.5 Combine like terms: -2.872281323 + 4.5 = 1.627718677 z = 1.627718677 Simplifying z = 1.627718677

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.372281323, 1.627718677}

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