8w^2+w+1=0

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Solution for 8w^2+w+1=0 equation:


Simplifying
8w2 + w + 1 = 0

Reorder the terms:
1 + w + 8w2 = 0

Solving
1 + w + 8w2 = 0

Solving for variable 'w'.

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

Divide each side by '8'.
0.125 + 0.125w + w2 = 0

Move the constant term to the right:

Add '-0.125' to each side of the equation.
0.125 + 0.125w + -0.125 + w2 = 0 + -0.125

Reorder the terms:
0.125 + -0.125 + 0.125w + w2 = 0 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + 0.125w + w2 = 0 + -0.125
0.125w + w2 = 0 + -0.125

Combine like terms: 0 + -0.125 = -0.125
0.125w + w2 = -0.125

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.
0.125w + 0.25 + w2 = -0.125 + 0.25

Reorder the terms:
0.25 + 0.125w + w2 = -0.125 + 0.25

Combine like terms: -0.125 + 0.25 = 0.125
0.25 + 0.125w + w2 = 0.125

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 0.125

Calculate the square root of the right side: 0.353553391

Break this problem into two subproblems by setting 
(w + 0.5) equal to 0.353553391 and -0.353553391.

Subproblem 1

w + 0.5 = 0.353553391 Simplifying w + 0.5 = 0.353553391 Reorder the terms: 0.5 + w = 0.353553391 Solving 0.5 + w = 0.353553391 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 = 0.353553391 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 0.353553391 + -0.5 w = 0.353553391 + -0.5 Combine like terms: 0.353553391 + -0.5 = -0.146446609 w = -0.146446609 Simplifying w = -0.146446609

Subproblem 2

w + 0.5 = -0.353553391 Simplifying w + 0.5 = -0.353553391 Reorder the terms: 0.5 + w = -0.353553391 Solving 0.5 + w = -0.353553391 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 = -0.353553391 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -0.353553391 + -0.5 w = -0.353553391 + -0.5 Combine like terms: -0.353553391 + -0.5 = -0.853553391 w = -0.853553391 Simplifying w = -0.853553391

Solution

The solution to the problem is based on the solutions from the subproblems. w = {-0.146446609, -0.853553391}

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