z^2-7z-27=0

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Solution for z^2-7z-27=0 equation:


Simplifying
z2 + -7z + -27 = 0

Reorder the terms:
-27 + -7z + z2 = 0

Solving
-27 + -7z + 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 + -7z + 27 + z2 = 0 + 27

Reorder the terms:
-27 + 27 + -7z + z2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + -7z + z2 = 0 + 27
-7z + z2 = 0 + 27

Combine like terms: 0 + 27 = 27
-7z + z2 = 27

The z term is -7z.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7z + 12.25 + z2 = 27 + 12.25

Reorder the terms:
12.25 + -7z + z2 = 27 + 12.25

Combine like terms: 27 + 12.25 = 39.25
12.25 + -7z + z2 = 39.25

Factor a perfect square on the left side:
(z + -3.5)(z + -3.5) = 39.25

Calculate the square root of the right side: 6.264982043

Break this problem into two subproblems by setting 
(z + -3.5) equal to 6.264982043 and -6.264982043.

Subproblem 1

z + -3.5 = 6.264982043 Simplifying z + -3.5 = 6.264982043 Reorder the terms: -3.5 + z = 6.264982043 Solving -3.5 + z = 6.264982043 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + z = 6.264982043 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = 6.264982043 + 3.5 z = 6.264982043 + 3.5 Combine like terms: 6.264982043 + 3.5 = 9.764982043 z = 9.764982043 Simplifying z = 9.764982043

Subproblem 2

z + -3.5 = -6.264982043 Simplifying z + -3.5 = -6.264982043 Reorder the terms: -3.5 + z = -6.264982043 Solving -3.5 + z = -6.264982043 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + z = -6.264982043 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = -6.264982043 + 3.5 z = -6.264982043 + 3.5 Combine like terms: -6.264982043 + 3.5 = -2.764982043 z = -2.764982043 Simplifying z = -2.764982043

Solution

The solution to the problem is based on the solutions from the subproblems. z = {9.764982043, -2.764982043}

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