w^2-w-21=0

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Solution for w^2-w-21=0 equation: 


Simplifying
w2 + -1w + -21 = 0

Reorder the terms:
-21 + -1w + w2 = 0

Solving
-21 + -1w + w2 = 0

Solving for variable 'w'.

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -1w + 21 + w2 = 0 + 21

Reorder the terms:
-21 + 21 + -1w + w2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -1w + w2 = 0 + 21
-1w + w2 = 0 + 21

Combine like terms: 0 + 21 = 21
-1w + w2 = 21

The w term is -1w.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1w + 0.25 + w2 = 21 + 0.25

Reorder the terms:
0.25 + -1w + w2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + -1w + w2 = 21.25

Factor a perfect square on the left side:
(w + -0.5)(w + -0.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(w + -0.5) equal to 4.609772229 and -4.609772229.


Subproblem 1
w + -0.5 = 4.609772229

Simplifying
w + -0.5 = 4.609772229

Reorder the terms:
-0.5 + w = 4.609772229

Solving
-0.5 + w = 4.609772229

Solving for variable 'w'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + w = 4.609772229 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + w = 4.609772229 + 0.5
w = 4.609772229 + 0.5

Combine like terms: 4.609772229 + 0.5 = 5.109772229
w = 5.109772229

Simplifying
w = 5.109772229


Subproblem 2
w + -0.5 = -4.609772229

Simplifying
w + -0.5 = -4.609772229

Reorder the terms:
-0.5 + w = -4.609772229

Solving
-0.5 + w = -4.609772229

Solving for variable 'w'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + w = -4.609772229 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + w = -4.609772229 + 0.5
w = -4.609772229 + 0.5

Combine like terms: -4.609772229 + 0.5 = -4.109772229
w = -4.109772229

Simplifying
w = -4.109772229


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {5.109772229, -4.109772229}

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