4w^2+40w+1=9

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Solution for 4w^2+40w+1=9 equation: 


Simplifying
4w2 + 40w + 1 = 9

Reorder the terms:
1 + 40w + 4w2 = 9

Solving
1 + 40w + 4w2 = 9

Solving for variable 'w'.

Reorder the terms:
1 + -9 + 40w + 4w2 = 9 + -9

Combine like terms: 1 + -9 = -8
-8 + 40w + 4w2 = 9 + -9

Combine like terms: 9 + -9 = 0
-8 + 40w + 4w2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-2 + 10w + w2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-2 + 10w + w2)' equal to zero and attempt to solve:

Simplifying
-2 + 10w + w2 = 0

Solving
-2 + 10w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 10w + 2 + w2 = 0 + 2

Reorder the terms:
-2 + 2 + 10w + w2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 10w + w2 = 0 + 2
10w + w2 = 0 + 2

Combine like terms: 0 + 2 = 2
10w + w2 = 2

The w term is 10w.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10w + 25 + w2 = 2 + 25

Reorder the terms:
25 + 10w + w2 = 2 + 25

Combine like terms: 2 + 25 = 27
25 + 10w + w2 = 27

Factor a perfect square on the left side:
(w + 5)(w + 5) = 27

Calculate the square root of the right side: 5.196152423

Break this problem into two subproblems by setting 
(w + 5) equal to 5.196152423 and -5.196152423.


Subproblem 1
w + 5 = 5.196152423

Simplifying
w + 5 = 5.196152423

Reorder the terms:
5 + w = 5.196152423

Solving
5 + w = 5.196152423

Solving for variable 'w'.

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

Add '-5' to each side of the equation.
5 + -5 + w = 5.196152423 + -5

Combine like terms: 5 + -5 = 0
0 + w = 5.196152423 + -5
w = 5.196152423 + -5

Combine like terms: 5.196152423 + -5 = 0.196152423
w = 0.196152423

Simplifying
w = 0.196152423


Subproblem 2
w + 5 = -5.196152423

Simplifying
w + 5 = -5.196152423

Reorder the terms:
5 + w = -5.196152423

Solving
5 + w = -5.196152423

Solving for variable 'w'.

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

Add '-5' to each side of the equation.
5 + -5 + w = -5.196152423 + -5

Combine like terms: 5 + -5 = 0
0 + w = -5.196152423 + -5
w = -5.196152423 + -5

Combine like terms: -5.196152423 + -5 = -10.196152423
w = -10.196152423

Simplifying
w = -10.196152423


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.196152423, -10.196152423}
Solution
w = {0.196152423, -10.196152423}

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