24x^2+20x+1=0

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Solution for 24x^2+20x+1=0 equation: 


Simplifying
24x2 + 20x + 1 = 0

Reorder the terms:
1 + 20x + 24x2 = 0

Solving
1 + 20x + 24x2 = 0

Solving for variable 'x'.

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

Divide each side by '24'.
0.04166666667 + 0.8333333333x + x2 = 0

Move the constant term to the right:

Add '-0.04166666667' to each side of the equation.
0.04166666667 + 0.8333333333x + -0.04166666667 + x2 = 0 + -0.04166666667

Reorder the terms:
0.04166666667 + -0.04166666667 + 0.8333333333x + x2 = 0 + -0.04166666667

Combine like terms: 0.04166666667 + -0.04166666667 = 0.00000000000
0.00000000000 + 0.8333333333x + x2 = 0 + -0.04166666667
0.8333333333x + x2 = 0 + -0.04166666667

Combine like terms: 0 + -0.04166666667 = -0.04166666667
0.8333333333x + x2 = -0.04166666667

The x term is 0.8333333333x.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333x + 0.1736111111 + x2 = -0.04166666667 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333x + x2 = -0.04166666667 + 0.1736111111

Combine like terms: -0.04166666667 + 0.1736111111 = 0.13194444443
0.1736111111 + 0.8333333333x + x2 = 0.13194444443

Factor a perfect square on the left side:
(x + 0.4166666667)(x + 0.4166666667) = 0.13194444443

Calculate the square root of the right side: 0.363241579

Break this problem into two subproblems by setting 
(x + 0.4166666667) equal to 0.363241579 and -0.363241579.


Subproblem 1
x + 0.4166666667 = 0.363241579

Simplifying
x + 0.4166666667 = 0.363241579

Reorder the terms:
0.4166666667 + x = 0.363241579

Solving
0.4166666667 + x = 0.363241579

Solving for variable 'x'.

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

Add '-0.4166666667' to each side of the equation.
0.4166666667 + -0.4166666667 + x = 0.363241579 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = 0.363241579 + -0.4166666667
x = 0.363241579 + -0.4166666667

Combine like terms: 0.363241579 + -0.4166666667 = -0.0534250877
x = -0.0534250877

Simplifying
x = -0.0534250877


Subproblem 2
x + 0.4166666667 = -0.363241579

Simplifying
x + 0.4166666667 = -0.363241579

Reorder the terms:
0.4166666667 + x = -0.363241579

Solving
0.4166666667 + x = -0.363241579

Solving for variable 'x'.

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

Add '-0.4166666667' to each side of the equation.
0.4166666667 + -0.4166666667 + x = -0.363241579 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = -0.363241579 + -0.4166666667
x = -0.363241579 + -0.4166666667

Combine like terms: -0.363241579 + -0.4166666667 = -0.7799082457
x = -0.7799082457

Simplifying
x = -0.7799082457


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.0534250877, -0.7799082457}

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