w(3w+8)=24

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Solution for w(3w+8)=24 equation: 


Simplifying
w(3w + 8) = 24

Reorder the terms:
w(8 + 3w) = 24
(8 * w + 3w * w) = 24
(8w + 3w2) = 24

Solving
8w + 3w2 = 24

Solving for variable 'w'.

Reorder the terms:
-24 + 8w + 3w2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 8w + 3w2 = 0

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

Divide each side by '3'.
-8 + 2.666666667w + w2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 2.666666667w + 8 + w2 = 0 + 8

Reorder the terms:
-8 + 8 + 2.666666667w + w2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 2.666666667w + w2 = 0 + 8
2.666666667w + w2 = 0 + 8

Combine like terms: 0 + 8 = 8
2.666666667w + w2 = 8

The w term is 2.666666667w.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667w + 1.777777780 + w2 = 8 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667w + w2 = 8 + 1.777777780

Combine like terms: 8 + 1.777777780 = 9.77777778
1.777777780 + 2.666666667w + w2 = 9.77777778

Factor a perfect square on the left side:
(w + 1.333333334)(w + 1.333333334) = 9.77777778

Calculate the square root of the right side: 3.12694384

Break this problem into two subproblems by setting 
(w + 1.333333334) equal to 3.12694384 and -3.12694384.


Subproblem 1
w + 1.333333334 = 3.12694384

Simplifying
w + 1.333333334 = 3.12694384

Reorder the terms:
1.333333334 + w = 3.12694384

Solving
1.333333334 + w = 3.12694384

Solving for variable 'w'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + w = 3.12694384 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + w = 3.12694384 + -1.333333334
w = 3.12694384 + -1.333333334

Combine like terms: 3.12694384 + -1.333333334 = 1.793610506
w = 1.793610506

Simplifying
w = 1.793610506


Subproblem 2
w + 1.333333334 = -3.12694384

Simplifying
w + 1.333333334 = -3.12694384

Reorder the terms:
1.333333334 + w = -3.12694384

Solving
1.333333334 + w = -3.12694384

Solving for variable 'w'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + w = -3.12694384 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + w = -3.12694384 + -1.333333334
w = -3.12694384 + -1.333333334

Combine like terms: -3.12694384 + -1.333333334 = -4.460277174
w = -4.460277174

Simplifying
w = -4.460277174


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {1.793610506, -4.460277174}

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