4x^2+6x-24=0

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Solution for 4x^2+6x-24=0 equation: 


Simplifying
4x2 + 6x + -24 = 0

Reorder the terms:
-24 + 6x + 4x2 = 0

Solving
-24 + 6x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-12 + 3x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-12 + 3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-12 + 3x + 2x2 = 0

Solving
-12 + 3x + 2x2 = 0

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

Divide each side by '2'.
-6 + 1.5x + x2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 1.5x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + 1.5x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 1.5x + x2 = 0 + 6
1.5x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
1.5x + x2 = 6

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 = 6 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 6 + 0.5625

Combine like terms: 6 + 0.5625 = 6.5625
0.5625 + 1.5x + x2 = 6.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(x + 0.75) equal to 2.561737691 and -2.561737691.


Subproblem 1
x + 0.75 = 2.561737691

Simplifying
x + 0.75 = 2.561737691

Reorder the terms:
0.75 + x = 2.561737691

Solving
0.75 + x = 2.561737691

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 = 2.561737691 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 2.561737691 + -0.75
x = 2.561737691 + -0.75

Combine like terms: 2.561737691 + -0.75 = 1.811737691
x = 1.811737691

Simplifying
x = 1.811737691


Subproblem 2
x + 0.75 = -2.561737691

Simplifying
x + 0.75 = -2.561737691

Reorder the terms:
0.75 + x = -2.561737691

Solving
0.75 + x = -2.561737691

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 = -2.561737691 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -2.561737691 + -0.75
x = -2.561737691 + -0.75

Combine like terms: -2.561737691 + -0.75 = -3.311737691
x = -3.311737691

Simplifying
x = -3.311737691


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.811737691, -3.311737691}
Solution
x = {1.811737691, -3.311737691}

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