24x^2+15x-3=0

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Solution for 24x^2+15x-3=0 equation: 


Simplifying
24x2 + 15x + -3 = 0

Reorder the terms:
-3 + 15x + 24x2 = 0

Solving
-3 + 15x + 24x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-1 + 5x + 8x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-1 + 5x + 8x2)' equal to zero and attempt to solve:

Simplifying
-1 + 5x + 8x2 = 0

Solving
-1 + 5x + 8x2 = 0

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

Divide each side by '8'.
-0.125 + 0.625x + x2 = 0

Move the constant term to the right:

Add '0.125' to each side of the equation.
-0.125 + 0.625x + 0.125 + x2 = 0 + 0.125

Reorder the terms:
-0.125 + 0.125 + 0.625x + x2 = 0 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + 0.625x + x2 = 0 + 0.125
0.625x + x2 = 0 + 0.125

Combine like terms: 0 + 0.125 = 0.125
0.625x + x2 = 0.125

The x term is 0.625x.  Take half its coefficient (0.3125).
Square it (0.09765625) and add it to both sides.

Add '0.09765625' to each side of the equation.
0.625x + 0.09765625 + x2 = 0.125 + 0.09765625

Reorder the terms:
0.09765625 + 0.625x + x2 = 0.125 + 0.09765625

Combine like terms: 0.125 + 0.09765625 = 0.22265625
0.09765625 + 0.625x + x2 = 0.22265625

Factor a perfect square on the left side:
(x + 0.3125)(x + 0.3125) = 0.22265625

Calculate the square root of the right side: 0.471864652

Break this problem into two subproblems by setting 
(x + 0.3125) equal to 0.471864652 and -0.471864652.


Subproblem 1
x + 0.3125 = 0.471864652

Simplifying
x + 0.3125 = 0.471864652

Reorder the terms:
0.3125 + x = 0.471864652

Solving
0.3125 + x = 0.471864652

Solving for variable 'x'.

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

Add '-0.3125' to each side of the equation.
0.3125 + -0.3125 + x = 0.471864652 + -0.3125

Combine like terms: 0.3125 + -0.3125 = 0.0000
0.0000 + x = 0.471864652 + -0.3125
x = 0.471864652 + -0.3125

Combine like terms: 0.471864652 + -0.3125 = 0.159364652
x = 0.159364652

Simplifying
x = 0.159364652


Subproblem 2
x + 0.3125 = -0.471864652

Simplifying
x + 0.3125 = -0.471864652

Reorder the terms:
0.3125 + x = -0.471864652

Solving
0.3125 + x = -0.471864652

Solving for variable 'x'.

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

Add '-0.3125' to each side of the equation.
0.3125 + -0.3125 + x = -0.471864652 + -0.3125

Combine like terms: 0.3125 + -0.3125 = 0.0000
0.0000 + x = -0.471864652 + -0.3125
x = -0.471864652 + -0.3125

Combine like terms: -0.471864652 + -0.3125 = -0.784364652
x = -0.784364652

Simplifying
x = -0.784364652


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.159364652, -0.784364652}
Solution
x = {0.159364652, -0.784364652}

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