z^2-5z-7=0

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Solution for z^2-5z-7=0 equation: 


Simplifying
z2 + -5z + -7 = 0

Reorder the terms:
-7 + -5z + z2 = 0

Solving
-7 + -5z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -5z + 7 + z2 = 0 + 7

Reorder the terms:
-7 + 7 + -5z + z2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -5z + z2 = 0 + 7
-5z + z2 = 0 + 7

Combine like terms: 0 + 7 = 7
-5z + z2 = 7

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 = 7 + 6.25

Reorder the terms:
6.25 + -5z + z2 = 7 + 6.25

Combine like terms: 7 + 6.25 = 13.25
6.25 + -5z + z2 = 13.25

Factor a perfect square on the left side:
(z + -2.5)(z + -2.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(z + -2.5) equal to 3.640054945 and -3.640054945.


Subproblem 1
z + -2.5 = 3.640054945

Simplifying
z + -2.5 = 3.640054945

Reorder the terms:
-2.5 + z = 3.640054945

Solving
-2.5 + z = 3.640054945

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 = 3.640054945 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + z = 3.640054945 + 2.5
z = 3.640054945 + 2.5

Combine like terms: 3.640054945 + 2.5 = 6.140054945
z = 6.140054945

Simplifying
z = 6.140054945


Subproblem 2
z + -2.5 = -3.640054945

Simplifying
z + -2.5 = -3.640054945

Reorder the terms:
-2.5 + z = -3.640054945

Solving
-2.5 + z = -3.640054945

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 = -3.640054945 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + z = -3.640054945 + 2.5
z = -3.640054945 + 2.5

Combine like terms: -3.640054945 + 2.5 = -1.140054945
z = -1.140054945

Simplifying
z = -1.140054945


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {6.140054945, -1.140054945}

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