z^2-12z=-9

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


Simplifying
z2 + -12z = -9

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

Solving
-12z + z2 = -9

Solving for variable 'z'.

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The z term is -12z.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12z + 36 + z2 = -9 + 36

Reorder the terms:
36 + -12z + z2 = -9 + 36

Combine like terms: -9 + 36 = 27
36 + -12z + z2 = 27

Factor a perfect square on the left side:
(z + -6)(z + -6) = 27

Calculate the square root of the right side: 5.196152423

Break this problem into two subproblems by setting 
(z + -6) equal to 5.196152423 and -5.196152423.

Subproblem 1

z + -6 = 5.196152423 Simplifying z + -6 = 5.196152423 Reorder the terms: -6 + z = 5.196152423 Solving -6 + z = 5.196152423 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + z = 5.196152423 + 6 Combine like terms: -6 + 6 = 0 0 + z = 5.196152423 + 6 z = 5.196152423 + 6 Combine like terms: 5.196152423 + 6 = 11.196152423 z = 11.196152423 Simplifying z = 11.196152423

Subproblem 2

z + -6 = -5.196152423 Simplifying z + -6 = -5.196152423 Reorder the terms: -6 + z = -5.196152423 Solving -6 + z = -5.196152423 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + z = -5.196152423 + 6 Combine like terms: -6 + 6 = 0 0 + z = -5.196152423 + 6 z = -5.196152423 + 6 Combine like terms: -5.196152423 + 6 = 0.803847577 z = 0.803847577 Simplifying z = 0.803847577

Solution

The solution to the problem is based on the solutions from the subproblems. z = {11.196152423, 0.803847577}

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