7w-4w(w-9)=45

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Solution for 7w-4w(w-9)=45 equation: 


Simplifying
7w + -4w(w + -9) = 45

Reorder the terms:
7w + -4w(-9 + w) = 45
7w + (-9 * -4w + w * -4w) = 45
7w + (36w + -4w2) = 45

Combine like terms: 7w + 36w = 43w
43w + -4w2 = 45

Solving
43w + -4w2 = 45

Solving for variable 'w'.

Reorder the terms:
-45 + 43w + -4w2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 43w + -4w2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
11.25 + -10.75w + w2 = 0

Move the constant term to the right:

Add '-11.25' to each side of the equation.
11.25 + -10.75w + -11.25 + w2 = 0 + -11.25

Reorder the terms:
11.25 + -11.25 + -10.75w + w2 = 0 + -11.25

Combine like terms: 11.25 + -11.25 = 0.00
0.00 + -10.75w + w2 = 0 + -11.25
-10.75w + w2 = 0 + -11.25

Combine like terms: 0 + -11.25 = -11.25
-10.75w + w2 = -11.25

The w term is -10.75w.  Take half its coefficient (-5.375).
Square it (28.890625) and add it to both sides.

Add '28.890625' to each side of the equation.
-10.75w + 28.890625 + w2 = -11.25 + 28.890625

Reorder the terms:
28.890625 + -10.75w + w2 = -11.25 + 28.890625

Combine like terms: -11.25 + 28.890625 = 17.640625
28.890625 + -10.75w + w2 = 17.640625

Factor a perfect square on the left side:
(w + -5.375)(w + -5.375) = 17.640625

Calculate the square root of the right side: 4.200074404

Break this problem into two subproblems by setting 
(w + -5.375) equal to 4.200074404 and -4.200074404.


Subproblem 1
w + -5.375 = 4.200074404

Simplifying
w + -5.375 = 4.200074404

Reorder the terms:
-5.375 + w = 4.200074404

Solving
-5.375 + w = 4.200074404

Solving for variable 'w'.

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

Add '5.375' to each side of the equation.
-5.375 + 5.375 + w = 4.200074404 + 5.375

Combine like terms: -5.375 + 5.375 = 0.000
0.000 + w = 4.200074404 + 5.375
w = 4.200074404 + 5.375

Combine like terms: 4.200074404 + 5.375 = 9.575074404
w = 9.575074404

Simplifying
w = 9.575074404


Subproblem 2
w + -5.375 = -4.200074404

Simplifying
w + -5.375 = -4.200074404

Reorder the terms:
-5.375 + w = -4.200074404

Solving
-5.375 + w = -4.200074404

Solving for variable 'w'.

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

Add '5.375' to each side of the equation.
-5.375 + 5.375 + w = -4.200074404 + 5.375

Combine like terms: -5.375 + 5.375 = 0.000
0.000 + w = -4.200074404 + 5.375
w = -4.200074404 + 5.375

Combine like terms: -4.200074404 + 5.375 = 1.174925596
w = 1.174925596

Simplifying
w = 1.174925596


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {9.575074404, 1.174925596}

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