z^2-5z-27=0

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Solution for z^2-5z-27=0 equation:


Simplifying
z2 + -5z + -27 = 0

Reorder the terms:
-27 + -5z + z2 = 0

Solving
-27 + -5z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '27' to each side of the equation.
-27 + -5z + 27 + z2 = 0 + 27

Reorder the terms:
-27 + 27 + -5z + z2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + -5z + z2 = 0 + 27
-5z + z2 = 0 + 27

Combine like terms: 0 + 27 = 27
-5z + z2 = 27

The z term is -5z.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5z + 6.25 + z2 = 27 + 6.25

Reorder the terms:
6.25 + -5z + z2 = 27 + 6.25

Combine like terms: 27 + 6.25 = 33.25
6.25 + -5z + z2 = 33.25

Factor a perfect square on the left side:
(z + -2.5)(z + -2.5) = 33.25

Calculate the square root of the right side: 5.766281297

Break this problem into two subproblems by setting 
(z + -2.5) equal to 5.766281297 and -5.766281297.

Subproblem 1

z + -2.5 = 5.766281297 Simplifying z + -2.5 = 5.766281297 Reorder the terms: -2.5 + z = 5.766281297 Solving -2.5 + z = 5.766281297 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = 5.766281297 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 5.766281297 + 2.5 z = 5.766281297 + 2.5 Combine like terms: 5.766281297 + 2.5 = 8.266281297 z = 8.266281297 Simplifying z = 8.266281297

Subproblem 2

z + -2.5 = -5.766281297 Simplifying z + -2.5 = -5.766281297 Reorder the terms: -2.5 + z = -5.766281297 Solving -2.5 + z = -5.766281297 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = -5.766281297 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -5.766281297 + 2.5 z = -5.766281297 + 2.5 Combine like terms: -5.766281297 + 2.5 = -3.266281297 z = -3.266281297 Simplifying z = -3.266281297

Solution

The solution to the problem is based on the solutions from the subproblems. z = {8.266281297, -3.266281297}

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