6x^2-4x-224=0

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Solution for 6x^2-4x-224=0 equation: 


Simplifying
6x2 + -4x + -224 = 0

Reorder the terms:
-224 + -4x + 6x2 = 0

Solving
-224 + -4x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-112 + -2x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-112 + -2x + 3x2)' equal to zero and attempt to solve:

Simplifying
-112 + -2x + 3x2 = 0

Solving
-112 + -2x + 3x2 = 0

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

Divide each side by '3'.
-37.33333333 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '37.33333333' to each side of the equation.
-37.33333333 + -0.6666666667x + 37.33333333 + x2 = 0 + 37.33333333

Reorder the terms:
-37.33333333 + 37.33333333 + -0.6666666667x + x2 = 0 + 37.33333333

Combine like terms: -37.33333333 + 37.33333333 = 0.00000000
0.00000000 + -0.6666666667x + x2 = 0 + 37.33333333
-0.6666666667x + x2 = 0 + 37.33333333

Combine like terms: 0 + 37.33333333 = 37.33333333
-0.6666666667x + x2 = 37.33333333

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 37.33333333 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 37.33333333 + 0.1111111112

Combine like terms: 37.33333333 + 0.1111111112 = 37.4444444412
0.1111111112 + -0.6666666667x + x2 = 37.4444444412

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 37.4444444412

Calculate the square root of the right side: 6.119186583

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 6.119186583 and -6.119186583.


Subproblem 1
x + -0.3333333334 = 6.119186583

Simplifying
x + -0.3333333334 = 6.119186583

Reorder the terms:
-0.3333333334 + x = 6.119186583

Solving
-0.3333333334 + x = 6.119186583

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = 6.119186583 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 6.119186583 + 0.3333333334
x = 6.119186583 + 0.3333333334

Combine like terms: 6.119186583 + 0.3333333334 = 6.4525199164
x = 6.4525199164

Simplifying
x = 6.4525199164


Subproblem 2
x + -0.3333333334 = -6.119186583

Simplifying
x + -0.3333333334 = -6.119186583

Reorder the terms:
-0.3333333334 + x = -6.119186583

Solving
-0.3333333334 + x = -6.119186583

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = -6.119186583 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -6.119186583 + 0.3333333334
x = -6.119186583 + 0.3333333334

Combine like terms: -6.119186583 + 0.3333333334 = -5.7858532496
x = -5.7858532496

Simplifying
x = -5.7858532496


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.4525199164, -5.7858532496}
Solution
x = {6.4525199164, -5.7858532496}

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