24x^2-30x-240=0

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Solution for 24x^2-30x-240=0 equation: 


Simplifying
24x2 + -30x + -240 = 0

Reorder the terms:
-240 + -30x + 24x2 = 0

Solving
-240 + -30x + 24x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(-40 + -5x + 4x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-40 + -5x + 4x2)' equal to zero and attempt to solve:

Simplifying
-40 + -5x + 4x2 = 0

Solving
-40 + -5x + 4x2 = 0

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

Divide each side by '4'.
-10 + -1.25x + x2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -1.25x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + -1.25x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -1.25x + x2 = 0 + 10
-1.25x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
-1.25x + x2 = 10

The x term is -1.25x.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25x + 0.390625 + x2 = 10 + 0.390625

Reorder the terms:
0.390625 + -1.25x + x2 = 10 + 0.390625

Combine like terms: 10 + 0.390625 = 10.390625
0.390625 + -1.25x + x2 = 10.390625

Factor a perfect square on the left side:
(x + -0.625)(x + -0.625) = 10.390625

Calculate the square root of the right side: 3.22344924

Break this problem into two subproblems by setting 
(x + -0.625) equal to 3.22344924 and -3.22344924.


Subproblem 1
x + -0.625 = 3.22344924

Simplifying
x + -0.625 = 3.22344924

Reorder the terms:
-0.625 + x = 3.22344924

Solving
-0.625 + x = 3.22344924

Solving for variable 'x'.

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

Add '0.625' to each side of the equation.
-0.625 + 0.625 + x = 3.22344924 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + x = 3.22344924 + 0.625
x = 3.22344924 + 0.625

Combine like terms: 3.22344924 + 0.625 = 3.84844924
x = 3.84844924

Simplifying
x = 3.84844924


Subproblem 2
x + -0.625 = -3.22344924

Simplifying
x + -0.625 = -3.22344924

Reorder the terms:
-0.625 + x = -3.22344924

Solving
-0.625 + x = -3.22344924

Solving for variable 'x'.

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

Add '0.625' to each side of the equation.
-0.625 + 0.625 + x = -3.22344924 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + x = -3.22344924 + 0.625
x = -3.22344924 + 0.625

Combine like terms: -3.22344924 + 0.625 = -2.59844924
x = -2.59844924

Simplifying
x = -2.59844924


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.84844924, -2.59844924}
Solution
x = {3.84844924, -2.59844924}

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