20-4(2w+1)=9w(9+w)

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Solution for 20-4(2w+1)=9w(9+w) equation:


Simplifying
20 + -4(2w + 1) = 9w(9 + w)

Reorder the terms:
20 + -4(1 + 2w) = 9w(9 + w)
20 + (1 * -4 + 2w * -4) = 9w(9 + w)
20 + (-4 + -8w) = 9w(9 + w)

Combine like terms: 20 + -4 = 16
16 + -8w = 9w(9 + w)
16 + -8w = (9 * 9w + w * 9w)
16 + -8w = (81w + 9w2)

Solving
16 + -8w = 81w + 9w2

Solving for variable 'w'.

Combine like terms: -8w + -81w = -89w
16 + -89w + -9w2 = 81w + 9w2 + -81w + -9w2

Reorder the terms:
16 + -89w + -9w2 = 81w + -81w + 9w2 + -9w2

Combine like terms: 81w + -81w = 0
16 + -89w + -9w2 = 0 + 9w2 + -9w2
16 + -89w + -9w2 = 9w2 + -9w2

Combine like terms: 9w2 + -9w2 = 0
16 + -89w + -9w2 = 0

Begin completing the square.  Divide all terms by
-9 the coefficient of the squared term: 

Divide each side by '-9'.
-1.777777778 + 9.888888889w + w2 = 0

Move the constant term to the right:

Add '1.777777778' to each side of the equation.
-1.777777778 + 9.888888889w + 1.777777778 + w2 = 0 + 1.777777778

Reorder the terms:
-1.777777778 + 1.777777778 + 9.888888889w + w2 = 0 + 1.777777778

Combine like terms: -1.777777778 + 1.777777778 = 0.000000000
0.000000000 + 9.888888889w + w2 = 0 + 1.777777778
9.888888889w + w2 = 0 + 1.777777778

Combine like terms: 0 + 1.777777778 = 1.777777778
9.888888889w + w2 = 1.777777778

The w term is 9.888888889w.  Take half its coefficient (4.944444445).
Square it (24.44753087) and add it to both sides.

Add '24.44753087' to each side of the equation.
9.888888889w + 24.44753087 + w2 = 1.777777778 + 24.44753087

Reorder the terms:
24.44753087 + 9.888888889w + w2 = 1.777777778 + 24.44753087

Combine like terms: 1.777777778 + 24.44753087 = 26.225308648
24.44753087 + 9.888888889w + w2 = 26.225308648

Factor a perfect square on the left side:
(w + 4.944444445)(w + 4.944444445) = 26.225308648

Calculate the square root of the right side: 5.121065187

Break this problem into two subproblems by setting 
(w + 4.944444445) equal to 5.121065187 and -5.121065187.

Subproblem 1

w + 4.944444445 = 5.121065187 Simplifying w + 4.944444445 = 5.121065187 Reorder the terms: 4.944444445 + w = 5.121065187 Solving 4.944444445 + w = 5.121065187 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-4.944444445' to each side of the equation. 4.944444445 + -4.944444445 + w = 5.121065187 + -4.944444445 Combine like terms: 4.944444445 + -4.944444445 = 0.000000000 0.000000000 + w = 5.121065187 + -4.944444445 w = 5.121065187 + -4.944444445 Combine like terms: 5.121065187 + -4.944444445 = 0.176620742 w = 0.176620742 Simplifying w = 0.176620742

Subproblem 2

w + 4.944444445 = -5.121065187 Simplifying w + 4.944444445 = -5.121065187 Reorder the terms: 4.944444445 + w = -5.121065187 Solving 4.944444445 + w = -5.121065187 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-4.944444445' to each side of the equation. 4.944444445 + -4.944444445 + w = -5.121065187 + -4.944444445 Combine like terms: 4.944444445 + -4.944444445 = 0.000000000 0.000000000 + w = -5.121065187 + -4.944444445 w = -5.121065187 + -4.944444445 Combine like terms: -5.121065187 + -4.944444445 = -10.065509632 w = -10.065509632 Simplifying w = -10.065509632

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.176620742, -10.065509632}

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