3w^2+54w=75

Solution for 3w^2+54w=75 equation:

Simplifying
3w2 + 54w = 75

Reorder the terms:
54w + 3w2 = 75

Solving
54w + 3w2 = 75

Solving for variable 'w'.

Reorder the terms:
-75 + 54w + 3w2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + 54w + 3w2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-25 + 18w + w2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-25 + 18w + w2)' equal to zero and attempt to solve:

Simplifying
-25 + 18w + w2 = 0

Solving
-25 + 18w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + 18w + 25 + w2 = 0 + 25

Reorder the terms:
-25 + 25 + 18w + w2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + 18w + w2 = 0 + 25
18w + w2 = 0 + 25

Combine like terms: 0 + 25 = 25
18w + w2 = 25

The w term is 18w.  Take half its coefficient (9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
18w + 81 + w2 = 25 + 81

Reorder the terms:
81 + 18w + w2 = 25 + 81

Combine like terms: 25 + 81 = 106
81 + 18w + w2 = 106

Factor a perfect square on the left side:
(w + 9)(w + 9) = 106

Calculate the square root of the right side: 10.295630141

Break this problem into two subproblems by setting 
(w + 9) equal to 10.295630141 and -10.295630141.

Subproblem 1
w + 9 = 10.295630141

Simplifying
w + 9 = 10.295630141

Reorder the terms:
9 + w = 10.295630141

Solving
9 + w = 10.295630141

Solving for variable 'w'.

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

Add '-9' to each side of the equation.
9 + -9 + w = 10.295630141 + -9

Combine like terms: 9 + -9 = 0
0 + w = 10.295630141 + -9
w = 10.295630141 + -9

Combine like terms: 10.295630141 + -9 = 1.295630141
w = 1.295630141

Simplifying
w = 1.295630141

Subproblem 2
w + 9 = -10.295630141

Simplifying
w + 9 = -10.295630141

Reorder the terms:
9 + w = -10.295630141

Solving
9 + w = -10.295630141

Solving for variable 'w'.

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

Add '-9' to each side of the equation.
9 + -9 + w = -10.295630141 + -9

Combine like terms: 9 + -9 = 0
0 + w = -10.295630141 + -9
w = -10.295630141 + -9

Combine like terms: -10.295630141 + -9 = -19.295630141
w = -19.295630141

Simplifying
w = -19.295630141

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {1.295630141, -19.295630141}
Solution
w = {1.295630141, -19.295630141}