12w^2+28w+5=0

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Solution for 12w^2+28w+5=0 equation: 


Simplifying
12w2 + 28w + 5 = 0

Reorder the terms:
5 + 28w + 12w2 = 0

Solving
5 + 28w + 12w2 = 0

Solving for variable 'w'.

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

Divide each side by '12'.
0.4166666667 + 2.333333333w + w2 = 0

Move the constant term to the right:

Add '-0.4166666667' to each side of the equation.
0.4166666667 + 2.333333333w + -0.4166666667 + w2 = 0 + -0.4166666667

Reorder the terms:
0.4166666667 + -0.4166666667 + 2.333333333w + w2 = 0 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + 2.333333333w + w2 = 0 + -0.4166666667
2.333333333w + w2 = 0 + -0.4166666667

Combine like terms: 0 + -0.4166666667 = -0.4166666667
2.333333333w + w2 = -0.4166666667

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 = -0.4166666667 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333w + w2 = -0.4166666667 + 1.361111112

Combine like terms: -0.4166666667 + 1.361111112 = 0.9444444453
1.361111112 + 2.333333333w + w2 = 0.9444444453

Factor a perfect square on the left side:
(w + 1.166666667)(w + 1.166666667) = 0.9444444453

Calculate the square root of the right side: 0.971825316

Break this problem into two subproblems by setting 
(w + 1.166666667) equal to 0.971825316 and -0.971825316.


Subproblem 1
w + 1.166666667 = 0.971825316

Simplifying
w + 1.166666667 = 0.971825316

Reorder the terms:
1.166666667 + w = 0.971825316

Solving
1.166666667 + w = 0.971825316

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 = 0.971825316 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + w = 0.971825316 + -1.166666667
w = 0.971825316 + -1.166666667

Combine like terms: 0.971825316 + -1.166666667 = -0.194841351
w = -0.194841351

Simplifying
w = -0.194841351


Subproblem 2
w + 1.166666667 = -0.971825316

Simplifying
w + 1.166666667 = -0.971825316

Reorder the terms:
1.166666667 + w = -0.971825316

Solving
1.166666667 + w = -0.971825316

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 = -0.971825316 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + w = -0.971825316 + -1.166666667
w = -0.971825316 + -1.166666667

Combine like terms: -0.971825316 + -1.166666667 = -2.138491983
w = -2.138491983

Simplifying
w = -2.138491983


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-0.194841351, -2.138491983}

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