24a^2+12a-15=0

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Solution for 24a^2+12a-15=0 equation: 


Simplifying
24a2 + 12a + -15 = 0

Reorder the terms:
-15 + 12a + 24a2 = 0

Solving
-15 + 12a + 24a2 = 0

Solving for variable 'a'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-5 + 4a + 8a2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-5 + 4a + 8a2)' equal to zero and attempt to solve:

Simplifying
-5 + 4a + 8a2 = 0

Solving
-5 + 4a + 8a2 = 0

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

Divide each side by '8'.
-0.625 + 0.5a + a2 = 0

Move the constant term to the right:

Add '0.625' to each side of the equation.
-0.625 + 0.5a + 0.625 + a2 = 0 + 0.625

Reorder the terms:
-0.625 + 0.625 + 0.5a + a2 = 0 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + 0.5a + a2 = 0 + 0.625
0.5a + a2 = 0 + 0.625

Combine like terms: 0 + 0.625 = 0.625
0.5a + a2 = 0.625

The a term is 0.5a.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5a + 0.0625 + a2 = 0.625 + 0.0625

Reorder the terms:
0.0625 + 0.5a + a2 = 0.625 + 0.0625

Combine like terms: 0.625 + 0.0625 = 0.6875
0.0625 + 0.5a + a2 = 0.6875

Factor a perfect square on the left side:
(a + 0.25)(a + 0.25) = 0.6875

Calculate the square root of the right side: 0.829156198

Break this problem into two subproblems by setting 
(a + 0.25) equal to 0.829156198 and -0.829156198.


Subproblem 1
a + 0.25 = 0.829156198

Simplifying
a + 0.25 = 0.829156198

Reorder the terms:
0.25 + a = 0.829156198

Solving
0.25 + a = 0.829156198

Solving for variable 'a'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + a = 0.829156198 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + a = 0.829156198 + -0.25
a = 0.829156198 + -0.25

Combine like terms: 0.829156198 + -0.25 = 0.579156198
a = 0.579156198

Simplifying
a = 0.579156198


Subproblem 2
a + 0.25 = -0.829156198

Simplifying
a + 0.25 = -0.829156198

Reorder the terms:
0.25 + a = -0.829156198

Solving
0.25 + a = -0.829156198

Solving for variable 'a'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + a = -0.829156198 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + a = -0.829156198 + -0.25
a = -0.829156198 + -0.25

Combine like terms: -0.829156198 + -0.25 = -1.079156198
a = -1.079156198

Simplifying
a = -1.079156198


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.579156198, -1.079156198}
Solution
a = {0.579156198, -1.079156198}

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