3w^2+w=20

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Solution for 3w^2+w=20 equation:


Simplifying
3w2 + w = 20

Reorder the terms:
w + 3w2 = 20

Solving
w + 3w2 = 20

Solving for variable 'w'.

Reorder the terms:
-20 + w + 3w2 = 20 + -20

Combine like terms: 20 + -20 = 0
-20 + w + 3w2 = 0

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

Divide each side by '3'.
-6.666666667 + 0.3333333333w + w2 = 0

Move the constant term to the right:

Add '6.666666667' to each side of the equation.
-6.666666667 + 0.3333333333w + 6.666666667 + w2 = 0 + 6.666666667

Reorder the terms:
-6.666666667 + 6.666666667 + 0.3333333333w + w2 = 0 + 6.666666667

Combine like terms: -6.666666667 + 6.666666667 = 0.000000000
0.000000000 + 0.3333333333w + w2 = 0 + 6.666666667
0.3333333333w + w2 = 0 + 6.666666667

Combine like terms: 0 + 6.666666667 = 6.666666667
0.3333333333w + w2 = 6.666666667

The w term is w.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.3333333333w + 0.25 + w2 = 6.666666667 + 0.25

Reorder the terms:
0.25 + 0.3333333333w + w2 = 6.666666667 + 0.25

Combine like terms: 6.666666667 + 0.25 = 6.916666667
0.25 + 0.3333333333w + w2 = 6.916666667

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 6.916666667

Calculate the square root of the right side: 2.62995564

Break this problem into two subproblems by setting 
(w + 0.5) equal to 2.62995564 and -2.62995564.

Subproblem 1

w + 0.5 = 2.62995564 Simplifying w + 0.5 = 2.62995564 Reorder the terms: 0.5 + w = 2.62995564 Solving 0.5 + w = 2.62995564 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 2.62995564 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 2.62995564 + -0.5 w = 2.62995564 + -0.5 Combine like terms: 2.62995564 + -0.5 = 2.12995564 w = 2.12995564 Simplifying w = 2.12995564

Subproblem 2

w + 0.5 = -2.62995564 Simplifying w + 0.5 = -2.62995564 Reorder the terms: 0.5 + w = -2.62995564 Solving 0.5 + w = -2.62995564 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -2.62995564 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -2.62995564 + -0.5 w = -2.62995564 + -0.5 Combine like terms: -2.62995564 + -0.5 = -3.12995564 w = -3.12995564 Simplifying w = -3.12995564

Solution

The solution to the problem is based on the solutions from the subproblems. w = {2.12995564, -3.12995564}

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