z^2-6z-27=20

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Solution for z^2-6z-27=20 equation:


Simplifying
z2 + -6z + -27 = 20

Reorder the terms:
-27 + -6z + z2 = 20

Solving
-27 + -6z + z2 = 20

Solving for variable 'z'.

Reorder the terms:
-27 + -20 + -6z + z2 = 20 + -20

Combine like terms: -27 + -20 = -47
-47 + -6z + z2 = 20 + -20

Combine like terms: 20 + -20 = 0
-47 + -6z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '47' to each side of the equation.
-47 + -6z + 47 + z2 = 0 + 47

Reorder the terms:
-47 + 47 + -6z + z2 = 0 + 47

Combine like terms: -47 + 47 = 0
0 + -6z + z2 = 0 + 47
-6z + z2 = 0 + 47

Combine like terms: 0 + 47 = 47
-6z + z2 = 47

The z term is -6z.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6z + 9 + z2 = 47 + 9

Reorder the terms:
9 + -6z + z2 = 47 + 9

Combine like terms: 47 + 9 = 56
9 + -6z + z2 = 56

Factor a perfect square on the left side:
(z + -3)(z + -3) = 56

Calculate the square root of the right side: 7.483314774

Break this problem into two subproblems by setting 
(z + -3) equal to 7.483314774 and -7.483314774.

Subproblem 1

z + -3 = 7.483314774 Simplifying z + -3 = 7.483314774 Reorder the terms: -3 + z = 7.483314774 Solving -3 + z = 7.483314774 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 7.483314774 + 3 Combine like terms: -3 + 3 = 0 0 + z = 7.483314774 + 3 z = 7.483314774 + 3 Combine like terms: 7.483314774 + 3 = 10.483314774 z = 10.483314774 Simplifying z = 10.483314774

Subproblem 2

z + -3 = -7.483314774 Simplifying z + -3 = -7.483314774 Reorder the terms: -3 + z = -7.483314774 Solving -3 + z = -7.483314774 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -7.483314774 + 3 Combine like terms: -3 + 3 = 0 0 + z = -7.483314774 + 3 z = -7.483314774 + 3 Combine like terms: -7.483314774 + 3 = -4.483314774 z = -4.483314774 Simplifying z = -4.483314774

Solution

The solution to the problem is based on the solutions from the subproblems. z = {10.483314774, -4.483314774}

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