9w^2+24w+1=0

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Solution for 9w^2+24w+1=0 equation: 


Simplifying
9w2 + 24w + 1 = 0

Reorder the terms:
1 + 24w + 9w2 = 0

Solving
1 + 24w + 9w2 = 0

Solving for variable 'w'.

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

Divide each side by '9'.
0.1111111111 + 2.666666667w + w2 = 0

Move the constant term to the right:

Add '-0.1111111111' to each side of the equation.
0.1111111111 + 2.666666667w + -0.1111111111 + w2 = 0 + -0.1111111111

Reorder the terms:
0.1111111111 + -0.1111111111 + 2.666666667w + w2 = 0 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + 2.666666667w + w2 = 0 + -0.1111111111
2.666666667w + w2 = 0 + -0.1111111111

Combine like terms: 0 + -0.1111111111 = -0.1111111111
2.666666667w + w2 = -0.1111111111

The w term is 2.666666667w.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667w + 1.777777780 + w2 = -0.1111111111 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667w + w2 = -0.1111111111 + 1.777777780

Combine like terms: -0.1111111111 + 1.777777780 = 1.6666666689
1.777777780 + 2.666666667w + w2 = 1.6666666689

Factor a perfect square on the left side:
(w + 1.333333334)(w + 1.333333334) = 1.6666666689

Calculate the square root of the right side: 1.29099445

Break this problem into two subproblems by setting 
(w + 1.333333334) equal to 1.29099445 and -1.29099445.


Subproblem 1
w + 1.333333334 = 1.29099445

Simplifying
w + 1.333333334 = 1.29099445

Reorder the terms:
1.333333334 + w = 1.29099445

Solving
1.333333334 + w = 1.29099445

Solving for variable 'w'.

Move all terms containing w to the left, all other terms to the right.

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + w = 1.29099445 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + w = 1.29099445 + -1.333333334
w = 1.29099445 + -1.333333334

Combine like terms: 1.29099445 + -1.333333334 = -0.042338884
w = -0.042338884

Simplifying
w = -0.042338884


Subproblem 2
w + 1.333333334 = -1.29099445

Simplifying
w + 1.333333334 = -1.29099445

Reorder the terms:
1.333333334 + w = -1.29099445

Solving
1.333333334 + w = -1.29099445

Solving for variable 'w'.

Move all terms containing w to the left, all other terms to the right.

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + w = -1.29099445 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + w = -1.29099445 + -1.333333334
w = -1.29099445 + -1.333333334

Combine like terms: -1.29099445 + -1.333333334 = -2.624327784
w = -2.624327784

Simplifying
w = -2.624327784


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-0.042338884, -2.624327784}

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