24x^2+36x-45=0

Solution for 24x^2+36x-45=0 equation:

Simplifying
24x2 + 36x + -45 = 0

Reorder the terms:
-45 + 36x + 24x2 = 0

Solving
-45 + 36x + 24x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-15 + 12x + 8x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-15 + 12x + 8x2)' equal to zero and attempt to solve:

Simplifying
-15 + 12x + 8x2 = 0

Solving
-15 + 12x + 8x2 = 0

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

Divide each side by '8'.
-1.875 + 1.5x + x2 = 0

Move the constant term to the right:

Add '1.875' to each side of the equation.
-1.875 + 1.5x + 1.875 + x2 = 0 + 1.875

Reorder the terms:
-1.875 + 1.875 + 1.5x + x2 = 0 + 1.875

Combine like terms: -1.875 + 1.875 = 0.000
0.000 + 1.5x + x2 = 0 + 1.875
1.5x + x2 = 0 + 1.875

Combine like terms: 0 + 1.875 = 1.875
1.5x + x2 = 1.875

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 1.875 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 1.875 + 0.5625

Combine like terms: 1.875 + 0.5625 = 2.4375
0.5625 + 1.5x + x2 = 2.4375

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 2.4375

Calculate the square root of the right side: 1.5612495

Break this problem into two subproblems by setting 
(x + 0.75) equal to 1.5612495 and -1.5612495.

Subproblem 1
x + 0.75 = 1.5612495

Simplifying
x + 0.75 = 1.5612495

Reorder the terms:
0.75 + x = 1.5612495

Solving
0.75 + x = 1.5612495

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 1.5612495 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 1.5612495 + -0.75
x = 1.5612495 + -0.75

Combine like terms: 1.5612495 + -0.75 = 0.8112495
x = 0.8112495

Simplifying
x = 0.8112495

Subproblem 2
x + 0.75 = -1.5612495

Simplifying
x + 0.75 = -1.5612495

Reorder the terms:
0.75 + x = -1.5612495

Solving
0.75 + x = -1.5612495

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -1.5612495 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -1.5612495 + -0.75
x = -1.5612495 + -0.75

Combine like terms: -1.5612495 + -0.75 = -2.3112495
x = -2.3112495

Simplifying
x = -2.3112495

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.8112495, -2.3112495}
Solution
x = {0.8112495, -2.3112495}