(w-6)(w+6)+(5w-1)(2w+3)=0

Solution for (w-6)(w+6)+(5w-1)(2w+3)=0 equation:

Simplifying
(w + -6)(w + 6) + (5w + -1)(2w + 3) = 0

Reorder the terms:
(-6 + w)(w + 6) + (5w + -1)(2w + 3) = 0

Reorder the terms:
(-6 + w)(6 + w) + (5w + -1)(2w + 3) = 0

Multiply (-6 + w) * (6 + w)
(-6(6 + w) + w(6 + w)) + (5w + -1)(2w + 3) = 0
((6 * -6 + w * -6) + w(6 + w)) + (5w + -1)(2w + 3) = 0
((-36 + -6w) + w(6 + w)) + (5w + -1)(2w + 3) = 0
(-36 + -6w + (6 * w + w * w)) + (5w + -1)(2w + 3) = 0
(-36 + -6w + (6w + w2)) + (5w + -1)(2w + 3) = 0

Combine like terms: -6w + 6w = 0
(-36 + 0 + w2) + (5w + -1)(2w + 3) = 0
(-36 + w2) + (5w + -1)(2w + 3) = 0

Reorder the terms:
-36 + w2 + (-1 + 5w)(2w + 3) = 0

Reorder the terms:
-36 + w2 + (-1 + 5w)(3 + 2w) = 0

Multiply (-1 + 5w) * (3 + 2w)
-36 + w2 + (-1(3 + 2w) + 5w * (3 + 2w)) = 0
-36 + w2 + ((3 * -1 + 2w * -1) + 5w * (3 + 2w)) = 0
-36 + w2 + ((-3 + -2w) + 5w * (3 + 2w)) = 0
-36 + w2 + (-3 + -2w + (3 * 5w + 2w * 5w)) = 0
-36 + w2 + (-3 + -2w + (15w + 10w2)) = 0

Combine like terms: -2w + 15w = 13w
-36 + w2 + (-3 + 13w + 10w2) = 0

Reorder the terms:
-36 + -3 + 13w + w2 + 10w2 = 0

Combine like terms: -36 + -3 = -39
-39 + 13w + w2 + 10w2 = 0

Combine like terms: w2 + 10w2 = 11w2
-39 + 13w + 11w2 = 0

Solving
-39 + 13w + 11w2 = 0

Solving for variable 'w'.

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

Divide each side by '11'.
-3.545454545 + 1.181818182w + w2 = 0

Move the constant term to the right:

Add '3.545454545' to each side of the equation.
-3.545454545 + 1.181818182w + 3.545454545 + w2 = 0 + 3.545454545

Reorder the terms:
-3.545454545 + 3.545454545 + 1.181818182w + w2 = 0 + 3.545454545

Combine like terms: -3.545454545 + 3.545454545 = 0.000000000
0.000000000 + 1.181818182w + w2 = 0 + 3.545454545
1.181818182w + w2 = 0 + 3.545454545

Combine like terms: 0 + 3.545454545 = 3.545454545
1.181818182w + w2 = 3.545454545

The w term is 1.181818182w.  Take half its coefficient (0.590909091).
Square it (0.3491735538) and add it to both sides.

Add '0.3491735538' to each side of the equation.
1.181818182w + 0.3491735538 + w2 = 3.545454545 + 0.3491735538

Reorder the terms:
0.3491735538 + 1.181818182w + w2 = 3.545454545 + 0.3491735538

Combine like terms: 3.545454545 + 0.3491735538 = 3.8946280988
0.3491735538 + 1.181818182w + w2 = 3.8946280988

Factor a perfect square on the left side:
(w + 0.590909091)(w + 0.590909091) = 3.8946280988

Calculate the square root of the right side: 1.973481213

Break this problem into two subproblems by setting 
(w + 0.590909091) equal to 1.973481213 and -1.973481213.

Subproblem 1
w + 0.590909091 = 1.973481213

Simplifying
w + 0.590909091 = 1.973481213

Reorder the terms:
0.590909091 + w = 1.973481213

Solving
0.590909091 + w = 1.973481213

Solving for variable 'w'.

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

Add '-0.590909091' to each side of the equation.
0.590909091 + -0.590909091 + w = 1.973481213 + -0.590909091

Combine like terms: 0.590909091 + -0.590909091 = 0.000000000
0.000000000 + w = 1.973481213 + -0.590909091
w = 1.973481213 + -0.590909091

Combine like terms: 1.973481213 + -0.590909091 = 1.382572122
w = 1.382572122

Simplifying
w = 1.382572122

Subproblem 2
w + 0.590909091 = -1.973481213

Simplifying
w + 0.590909091 = -1.973481213

Reorder the terms:
0.590909091 + w = -1.973481213

Solving
0.590909091 + w = -1.973481213

Solving for variable 'w'.

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

Add '-0.590909091' to each side of the equation.
0.590909091 + -0.590909091 + w = -1.973481213 + -0.590909091

Combine like terms: 0.590909091 + -0.590909091 = 0.000000000
0.000000000 + w = -1.973481213 + -0.590909091
w = -1.973481213 + -0.590909091

Combine like terms: -1.973481213 + -0.590909091 = -2.564390304
w = -2.564390304

Simplifying
w = -2.564390304

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {1.382572122, -2.564390304}