4x^2+14x-6=0

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


Simplifying
4x2 + 14x + -6 = 0

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

Solving
-6 + 14x + 4x2 = 0

Solving for variable 'x'.

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

Ignore the factor 2.

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

Simplifying
-3 + 7x + 2x2 = 0

Solving
-3 + 7x + 2x2 = 0

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

Divide each side by '2'.
-1.5 + 3.5x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 3.5x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 3.5x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 3.5x + x2 = 0 + 1.5
3.5x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
3.5x + x2 = 1.5

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = 1.5 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = 1.5 + 3.0625

Combine like terms: 1.5 + 3.0625 = 4.5625
3.0625 + 3.5x + x2 = 4.5625

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 4.5625

Calculate the square root of the right side: 2.136000936

Break this problem into two subproblems by setting 
(x + 1.75) equal to 2.136000936 and -2.136000936.


Subproblem 1
x + 1.75 = 2.136000936

Simplifying
x + 1.75 = 2.136000936

Reorder the terms:
1.75 + x = 2.136000936

Solving
1.75 + x = 2.136000936

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = 2.136000936 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = 2.136000936 + -1.75
x = 2.136000936 + -1.75

Combine like terms: 2.136000936 + -1.75 = 0.386000936
x = 0.386000936

Simplifying
x = 0.386000936


Subproblem 2
x + 1.75 = -2.136000936

Simplifying
x + 1.75 = -2.136000936

Reorder the terms:
1.75 + x = -2.136000936

Solving
1.75 + x = -2.136000936

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = -2.136000936 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = -2.136000936 + -1.75
x = -2.136000936 + -1.75

Combine like terms: -2.136000936 + -1.75 = -3.886000936
x = -3.886000936

Simplifying
x = -3.886000936


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.386000936, -3.886000936}
Solution
x = {0.386000936, -3.886000936}

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