6x^2+14x-4=0

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Solution for 6x^2+14x-4=0 equation: 


Simplifying
6x2 + 14x + -4 = 0

Reorder the terms:
-4 + 14x + 6x2 = 0

Solving
-4 + 14x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-2 + 7x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-2 + 7x + 3x2)' equal to zero and attempt to solve:

Simplifying
-2 + 7x + 3x2 = 0

Solving
-2 + 7x + 3x2 = 0

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

Divide each side by '3'.
-0.6666666667 + 2.333333333x + x2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + 2.333333333x + 0.6666666667 + x2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + 2.333333333x + x2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + 2.333333333x + x2 = 0 + 0.6666666667
2.333333333x + x2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
2.333333333x + x2 = 0.6666666667

The x term is 2.333333333x.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333x + 1.361111112 + x2 = 0.6666666667 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333x + x2 = 0.6666666667 + 1.361111112

Combine like terms: 0.6666666667 + 1.361111112 = 2.0277777787
1.361111112 + 2.333333333x + x2 = 2.0277777787

Factor a perfect square on the left side:
(x + 1.166666667)(x + 1.166666667) = 2.0277777787

Calculate the square root of the right side: 1.424000625

Break this problem into two subproblems by setting 
(x + 1.166666667) equal to 1.424000625 and -1.424000625.


Subproblem 1
x + 1.166666667 = 1.424000625

Simplifying
x + 1.166666667 = 1.424000625

Reorder the terms:
1.166666667 + x = 1.424000625

Solving
1.166666667 + x = 1.424000625

Solving for variable 'x'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + x = 1.424000625 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + x = 1.424000625 + -1.166666667
x = 1.424000625 + -1.166666667

Combine like terms: 1.424000625 + -1.166666667 = 0.257333958
x = 0.257333958

Simplifying
x = 0.257333958


Subproblem 2
x + 1.166666667 = -1.424000625

Simplifying
x + 1.166666667 = -1.424000625

Reorder the terms:
1.166666667 + x = -1.424000625

Solving
1.166666667 + x = -1.424000625

Solving for variable 'x'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + x = -1.424000625 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + x = -1.424000625 + -1.166666667
x = -1.424000625 + -1.166666667

Combine like terms: -1.424000625 + -1.166666667 = -2.590667292
x = -2.590667292

Simplifying
x = -2.590667292


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.257333958, -2.590667292}
Solution
x = {0.257333958, -2.590667292}

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