3w^2-23w-13=0

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Solution for 3w^2-23w-13=0 equation: 


Simplifying
3w2 + -23w + -13 = 0

Reorder the terms:
-13 + -23w + 3w2 = 0

Solving
-13 + -23w + 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'.
-4.333333333 + -7.666666667w + w2 = 0

Move the constant term to the right:

Add '4.333333333' to each side of the equation.
-4.333333333 + -7.666666667w + 4.333333333 + w2 = 0 + 4.333333333

Reorder the terms:
-4.333333333 + 4.333333333 + -7.666666667w + w2 = 0 + 4.333333333

Combine like terms: -4.333333333 + 4.333333333 = 0.000000000
0.000000000 + -7.666666667w + w2 = 0 + 4.333333333
-7.666666667w + w2 = 0 + 4.333333333

Combine like terms: 0 + 4.333333333 = 4.333333333
-7.666666667w + w2 = 4.333333333

The w term is -7.666666667w.  Take half its coefficient (-3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
-7.666666667w + 14.69444445 + w2 = 4.333333333 + 14.69444445

Reorder the terms:
14.69444445 + -7.666666667w + w2 = 4.333333333 + 14.69444445

Combine like terms: 4.333333333 + 14.69444445 = 19.027777783
14.69444445 + -7.666666667w + w2 = 19.027777783

Factor a perfect square on the left side:
(w + -3.833333334)(w + -3.833333334) = 19.027777783

Calculate the square root of the right side: 4.36208411

Break this problem into two subproblems by setting 
(w + -3.833333334) equal to 4.36208411 and -4.36208411.


Subproblem 1
w + -3.833333334 = 4.36208411

Simplifying
w + -3.833333334 = 4.36208411

Reorder the terms:
-3.833333334 + w = 4.36208411

Solving
-3.833333334 + w = 4.36208411

Solving for variable 'w'.

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

Add '3.833333334' to each side of the equation.
-3.833333334 + 3.833333334 + w = 4.36208411 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + w = 4.36208411 + 3.833333334
w = 4.36208411 + 3.833333334

Combine like terms: 4.36208411 + 3.833333334 = 8.195417444
w = 8.195417444

Simplifying
w = 8.195417444


Subproblem 2
w + -3.833333334 = -4.36208411

Simplifying
w + -3.833333334 = -4.36208411

Reorder the terms:
-3.833333334 + w = -4.36208411

Solving
-3.833333334 + w = -4.36208411

Solving for variable 'w'.

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

Add '3.833333334' to each side of the equation.
-3.833333334 + 3.833333334 + w = -4.36208411 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + w = -4.36208411 + 3.833333334
w = -4.36208411 + 3.833333334

Combine like terms: -4.36208411 + 3.833333334 = -0.528750776
w = -0.528750776

Simplifying
w = -0.528750776


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {8.195417444, -0.528750776}

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