5w^2=31w-28

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Solution for 5w^2=31w-28 equation:


Simplifying
5w2 = 31w + -28

Reorder the terms:
5w2 = -28 + 31w

Solving
5w2 = -28 + 31w

Solving for variable 'w'.

Reorder the terms:
28 + -31w + 5w2 = -28 + 31w + 28 + -31w

Reorder the terms:
28 + -31w + 5w2 = -28 + 28 + 31w + -31w

Combine like terms: -28 + 28 = 0
28 + -31w + 5w2 = 0 + 31w + -31w
28 + -31w + 5w2 = 31w + -31w

Combine like terms: 31w + -31w = 0
28 + -31w + 5w2 = 0

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

Divide each side by '5'.
5.6 + -6.2w + w2 = 0

Move the constant term to the right:

Add '-5.6' to each side of the equation.
5.6 + -6.2w + -5.6 + w2 = 0 + -5.6

Reorder the terms:
5.6 + -5.6 + -6.2w + w2 = 0 + -5.6

Combine like terms: 5.6 + -5.6 = 0.0
0.0 + -6.2w + w2 = 0 + -5.6
-6.2w + w2 = 0 + -5.6

Combine like terms: 0 + -5.6 = -5.6
-6.2w + w2 = -5.6

The w term is -6.2w.  Take half its coefficient (-3.1).
Square it (9.61) and add it to both sides.

Add '9.61' to each side of the equation.
-6.2w + 9.61 + w2 = -5.6 + 9.61

Reorder the terms:
9.61 + -6.2w + w2 = -5.6 + 9.61

Combine like terms: -5.6 + 9.61 = 4.01
9.61 + -6.2w + w2 = 4.01

Factor a perfect square on the left side:
(w + -3.1)(w + -3.1) = 4.01

Calculate the square root of the right side: 2.002498439

Break this problem into two subproblems by setting 
(w + -3.1) equal to 2.002498439 and -2.002498439.

Subproblem 1

w + -3.1 = 2.002498439 Simplifying w + -3.1 = 2.002498439 Reorder the terms: -3.1 + w = 2.002498439 Solving -3.1 + w = 2.002498439 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.1' to each side of the equation. -3.1 + 3.1 + w = 2.002498439 + 3.1 Combine like terms: -3.1 + 3.1 = 0.0 0.0 + w = 2.002498439 + 3.1 w = 2.002498439 + 3.1 Combine like terms: 2.002498439 + 3.1 = 5.102498439 w = 5.102498439 Simplifying w = 5.102498439

Subproblem 2

w + -3.1 = -2.002498439 Simplifying w + -3.1 = -2.002498439 Reorder the terms: -3.1 + w = -2.002498439 Solving -3.1 + w = -2.002498439 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.1' to each side of the equation. -3.1 + 3.1 + w = -2.002498439 + 3.1 Combine like terms: -3.1 + 3.1 = 0.0 0.0 + w = -2.002498439 + 3.1 w = -2.002498439 + 3.1 Combine like terms: -2.002498439 + 3.1 = 1.097501561 w = 1.097501561 Simplifying w = 1.097501561

Solution

The solution to the problem is based on the solutions from the subproblems. w = {5.102498439, 1.097501561}

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