4w^2+12w-9=0

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Solution for 4w^2+12w-9=0 equation:


Simplifying
4w2 + 12w + -9 = 0

Reorder the terms:
-9 + 12w + 4w2 = 0

Solving
-9 + 12w + 4w2 = 0

Solving for variable 'w'.

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

Divide each side by '4'.
-2.25 + 3w + w2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + 3w + 2.25 + w2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + 3w + w2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + 3w + w2 = 0 + 2.25
3w + w2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
3w + w2 = 2.25

The w term is 3w.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3w + 2.25 + w2 = 2.25 + 2.25

Reorder the terms:
2.25 + 3w + w2 = 2.25 + 2.25

Combine like terms: 2.25 + 2.25 = 4.5
2.25 + 3w + w2 = 4.5

Factor a perfect square on the left side:
(w + 1.5)(w + 1.5) = 4.5

Calculate the square root of the right side: 2.121320344

Break this problem into two subproblems by setting 
(w + 1.5) equal to 2.121320344 and -2.121320344.

Subproblem 1

w + 1.5 = 2.121320344 Simplifying w + 1.5 = 2.121320344 Reorder the terms: 1.5 + w = 2.121320344 Solving 1.5 + w = 2.121320344 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 2.121320344 + -1.5 w = 2.121320344 + -1.5 Combine like terms: 2.121320344 + -1.5 = 0.621320344 w = 0.621320344 Simplifying w = 0.621320344

Subproblem 2

w + 1.5 = -2.121320344 Simplifying w + 1.5 = -2.121320344 Reorder the terms: 1.5 + w = -2.121320344 Solving 1.5 + w = -2.121320344 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -2.121320344 + -1.5 w = -2.121320344 + -1.5 Combine like terms: -2.121320344 + -1.5 = -3.621320344 w = -3.621320344 Simplifying w = -3.621320344

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.621320344, -3.621320344}

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