w^2-20w-23=0

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Solution for w^2-20w-23=0 equation: 


Simplifying
w2 + -20w + -23 = 0

Reorder the terms:
-23 + -20w + w2 = 0

Solving
-23 + -20w + w2 = 0

Solving for variable 'w'.

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + -20w + 23 + w2 = 0 + 23

Reorder the terms:
-23 + 23 + -20w + w2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + -20w + w2 = 0 + 23
-20w + w2 = 0 + 23

Combine like terms: 0 + 23 = 23
-20w + w2 = 23

The w term is -20w.  Take half its coefficient (-10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
-20w + 100 + w2 = 23 + 100

Reorder the terms:
100 + -20w + w2 = 23 + 100

Combine like terms: 23 + 100 = 123
100 + -20w + w2 = 123

Factor a perfect square on the left side:
(w + -10)(w + -10) = 123

Calculate the square root of the right side: 11.090536506

Break this problem into two subproblems by setting 
(w + -10) equal to 11.090536506 and -11.090536506.


Subproblem 1
w + -10 = 11.090536506

Simplifying
w + -10 = 11.090536506

Reorder the terms:
-10 + w = 11.090536506

Solving
-10 + w = 11.090536506

Solving for variable 'w'.

Move all terms containing w to the left, all other terms to the right.

Add '10' to each side of the equation.
-10 + 10 + w = 11.090536506 + 10

Combine like terms: -10 + 10 = 0
0 + w = 11.090536506 + 10
w = 11.090536506 + 10

Combine like terms: 11.090536506 + 10 = 21.090536506
w = 21.090536506

Simplifying
w = 21.090536506


Subproblem 2
w + -10 = -11.090536506

Simplifying
w + -10 = -11.090536506

Reorder the terms:
-10 + w = -11.090536506

Solving
-10 + w = -11.090536506

Solving for variable 'w'.

Move all terms containing w to the left, all other terms to the right.

Add '10' to each side of the equation.
-10 + 10 + w = -11.090536506 + 10

Combine like terms: -10 + 10 = 0
0 + w = -11.090536506 + 10
w = -11.090536506 + 10

Combine like terms: -11.090536506 + 10 = -1.090536506
w = -1.090536506

Simplifying
w = -1.090536506


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {21.090536506, -1.090536506}

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