3w^2-7w-120=0

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Solution for 3w^2-7w-120=0 equation: 


Simplifying
3w2 + -7w + -120 = 0

Reorder the terms:
-120 + -7w + 3w2 = 0

Solving
-120 + -7w + 3w2 = 0

Solving for variable 'w'.

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

Divide each side by '3'.
-40 + -2.333333333w + w2 = 0

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + -2.333333333w + 40 + w2 = 0 + 40

Reorder the terms:
-40 + 40 + -2.333333333w + w2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + -2.333333333w + w2 = 0 + 40
-2.333333333w + w2 = 0 + 40

Combine like terms: 0 + 40 = 40
-2.333333333w + w2 = 40

The w term is -2.333333333w.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333w + 1.361111112 + w2 = 40 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333w + w2 = 40 + 1.361111112

Combine like terms: 40 + 1.361111112 = 41.361111112
1.361111112 + -2.333333333w + w2 = 41.361111112

Factor a perfect square on the left side:
(w + -1.166666667)(w + -1.166666667) = 41.361111112

Calculate the square root of the right side: 6.431260461

Break this problem into two subproblems by setting 
(w + -1.166666667) equal to 6.431260461 and -6.431260461.


Subproblem 1
w + -1.166666667 = 6.431260461

Simplifying
w + -1.166666667 = 6.431260461

Reorder the terms:
-1.166666667 + w = 6.431260461

Solving
-1.166666667 + w = 6.431260461

Solving for variable 'w'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + w = 6.431260461 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + w = 6.431260461 + 1.166666667
w = 6.431260461 + 1.166666667

Combine like terms: 6.431260461 + 1.166666667 = 7.597927128
w = 7.597927128

Simplifying
w = 7.597927128


Subproblem 2
w + -1.166666667 = -6.431260461

Simplifying
w + -1.166666667 = -6.431260461

Reorder the terms:
-1.166666667 + w = -6.431260461

Solving
-1.166666667 + w = -6.431260461

Solving for variable 'w'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + w = -6.431260461 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + w = -6.431260461 + 1.166666667
w = -6.431260461 + 1.166666667

Combine like terms: -6.431260461 + 1.166666667 = -5.264593794
w = -5.264593794

Simplifying
w = -5.264593794


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {7.597927128, -5.264593794}

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