15w^2+31w-28=0

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Solution for 15w^2+31w-28=0 equation:


Simplifying
15w2 + 31w + -28 = 0

Reorder the terms:
-28 + 31w + 15w2 = 0

Solving
-28 + 31w + 15w2 = 0

Solving for variable 'w'.

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

Divide each side by '15'.
-1.866666667 + 2.066666667w + w2 = 0

Move the constant term to the right:

Add '1.866666667' to each side of the equation.
-1.866666667 + 2.066666667w + 1.866666667 + w2 = 0 + 1.866666667

Reorder the terms:
-1.866666667 + 1.866666667 + 2.066666667w + w2 = 0 + 1.866666667

Combine like terms: -1.866666667 + 1.866666667 = 0.000000000
0.000000000 + 2.066666667w + w2 = 0 + 1.866666667
2.066666667w + w2 = 0 + 1.866666667

Combine like terms: 0 + 1.866666667 = 1.866666667
2.066666667w + w2 = 1.866666667

The w term is 2.066666667w.  Take half its coefficient (1.033333334).
Square it (1.067777779) and add it to both sides.

Add '1.067777779' to each side of the equation.
2.066666667w + 1.067777779 + w2 = 1.866666667 + 1.067777779

Reorder the terms:
1.067777779 + 2.066666667w + w2 = 1.866666667 + 1.067777779

Combine like terms: 1.866666667 + 1.067777779 = 2.934444446
1.067777779 + 2.066666667w + w2 = 2.934444446

Factor a perfect square on the left side:
(w + 1.033333334)(w + 1.033333334) = 2.934444446

Calculate the square root of the right side: 1.713022021

Break this problem into two subproblems by setting 
(w + 1.033333334) equal to 1.713022021 and -1.713022021.

Subproblem 1

w + 1.033333334 = 1.713022021 Simplifying w + 1.033333334 = 1.713022021 Reorder the terms: 1.033333334 + w = 1.713022021 Solving 1.033333334 + w = 1.713022021 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + w = 1.713022021 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + w = 1.713022021 + -1.033333334 w = 1.713022021 + -1.033333334 Combine like terms: 1.713022021 + -1.033333334 = 0.679688687 w = 0.679688687 Simplifying w = 0.679688687

Subproblem 2

w + 1.033333334 = -1.713022021 Simplifying w + 1.033333334 = -1.713022021 Reorder the terms: 1.033333334 + w = -1.713022021 Solving 1.033333334 + w = -1.713022021 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + w = -1.713022021 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + w = -1.713022021 + -1.033333334 w = -1.713022021 + -1.033333334 Combine like terms: -1.713022021 + -1.033333334 = -2.746355355 w = -2.746355355 Simplifying w = -2.746355355

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.679688687, -2.746355355}

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