6w^2+9w+2=0

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Solution for 6w^2+9w+2=0 equation:


Simplifying
6w2 + 9w + 2 = 0

Reorder the terms:
2 + 9w + 6w2 = 0

Solving
2 + 9w + 6w2 = 0

Solving for variable 'w'.

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

Divide each side by '6'.
0.3333333333 + 1.5w + w2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 1.5w + -0.3333333333 + w2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 1.5w + w2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 1.5w + w2 = 0 + -0.3333333333
1.5w + w2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
1.5w + w2 = -0.3333333333

The w term is 1.5w.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5w + 0.5625 + w2 = -0.3333333333 + 0.5625

Reorder the terms:
0.5625 + 1.5w + w2 = -0.3333333333 + 0.5625

Combine like terms: -0.3333333333 + 0.5625 = 0.2291666667
0.5625 + 1.5w + w2 = 0.2291666667

Factor a perfect square on the left side:
(w + 0.75)(w + 0.75) = 0.2291666667

Calculate the square root of the right side: 0.478713554

Break this problem into two subproblems by setting 
(w + 0.75) equal to 0.478713554 and -0.478713554.

Subproblem 1

w + 0.75 = 0.478713554 Simplifying w + 0.75 = 0.478713554 Reorder the terms: 0.75 + w = 0.478713554 Solving 0.75 + w = 0.478713554 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = 0.478713554 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 0.478713554 + -0.75 w = 0.478713554 + -0.75 Combine like terms: 0.478713554 + -0.75 = -0.271286446 w = -0.271286446 Simplifying w = -0.271286446

Subproblem 2

w + 0.75 = -0.478713554 Simplifying w + 0.75 = -0.478713554 Reorder the terms: 0.75 + w = -0.478713554 Solving 0.75 + w = -0.478713554 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = -0.478713554 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -0.478713554 + -0.75 w = -0.478713554 + -0.75 Combine like terms: -0.478713554 + -0.75 = -1.228713554 w = -1.228713554 Simplifying w = -1.228713554

Solution

The solution to the problem is based on the solutions from the subproblems. w = {-0.271286446, -1.228713554}

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