3w-(w-5)(w-3)=2w+1

Solution for 3w-(w-5)(w-3)=2w+1 equation:

Simplifying
3w + -1(w + -5)(w + -3) = 2w + 1

Reorder the terms:
3w + -1(-5 + w)(w + -3) = 2w + 1

Reorder the terms:
3w + -1(-5 + w)(-3 + w) = 2w + 1

Multiply (-5 + w) * (-3 + w)
3w + -1(-5(-3 + w) + w(-3 + w)) = 2w + 1
3w + -1((-3 * -5 + w * -5) + w(-3 + w)) = 2w + 1
3w + -1((15 + -5w) + w(-3 + w)) = 2w + 1
3w + -1(15 + -5w + (-3 * w + w * w)) = 2w + 1
3w + -1(15 + -5w + (-3w + w2)) = 2w + 1

Combine like terms: -5w + -3w = -8w
3w + -1(15 + -8w + w2) = 2w + 1
3w + (15 * -1 + -8w * -1 + w2 * -1) = 2w + 1
3w + (-15 + 8w + -1w2) = 2w + 1

Reorder the terms:
-15 + 3w + 8w + -1w2 = 2w + 1

Combine like terms: 3w + 8w = 11w
-15 + 11w + -1w2 = 2w + 1

Reorder the terms:
-15 + 11w + -1w2 = 1 + 2w

Solving
-15 + 11w + -1w2 = 1 + 2w

Solving for variable 'w'.

Reorder the terms:
-15 + -1 + 11w + -2w + -1w2 = 1 + 2w + -1 + -2w

Combine like terms: -15 + -1 = -16
-16 + 11w + -2w + -1w2 = 1 + 2w + -1 + -2w

Combine like terms: 11w + -2w = 9w
-16 + 9w + -1w2 = 1 + 2w + -1 + -2w

Reorder the terms:
-16 + 9w + -1w2 = 1 + -1 + 2w + -2w

Combine like terms: 1 + -1 = 0
-16 + 9w + -1w2 = 0 + 2w + -2w
-16 + 9w + -1w2 = 2w + -2w

Combine like terms: 2w + -2w = 0
-16 + 9w + -1w2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
16 + -9w + w2 = 0

Move the constant term to the right:

Add '-16' to each side of the equation.
16 + -9w + -16 + w2 = 0 + -16

Reorder the terms:
16 + -16 + -9w + w2 = 0 + -16

Combine like terms: 16 + -16 = 0
0 + -9w + w2 = 0 + -16
-9w + w2 = 0 + -16

Combine like terms: 0 + -16 = -16
-9w + w2 = -16

The w term is -9w.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9w + 20.25 + w2 = -16 + 20.25

Reorder the terms:
20.25 + -9w + w2 = -16 + 20.25

Combine like terms: -16 + 20.25 = 4.25
20.25 + -9w + w2 = 4.25

Factor a perfect square on the left side:
(w + -4.5)(w + -4.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(w + -4.5) equal to 2.061552813 and -2.061552813.

Subproblem 1
w + -4.5 = 2.061552813

Simplifying
w + -4.5 = 2.061552813

Reorder the terms:
-4.5 + w = 2.061552813

Solving
-4.5 + w = 2.061552813

Solving for variable 'w'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + w = 2.061552813 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + w = 2.061552813 + 4.5
w = 2.061552813 + 4.5

Combine like terms: 2.061552813 + 4.5 = 6.561552813
w = 6.561552813

Simplifying
w = 6.561552813

Subproblem 2
w + -4.5 = -2.061552813

Simplifying
w + -4.5 = -2.061552813

Reorder the terms:
-4.5 + w = -2.061552813

Solving
-4.5 + w = -2.061552813

Solving for variable 'w'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + w = -2.061552813 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + w = -2.061552813 + 4.5
w = -2.061552813 + 4.5

Combine like terms: -2.061552813 + 4.5 = 2.438447187
w = 2.438447187

Simplifying
w = 2.438447187

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {6.561552813, 2.438447187}