6x^2-23x-14=0

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Solution for 6x^2-23x-14=0 equation: 


Simplifying
6x2 + -23x + -14 = 0

Reorder the terms:
-14 + -23x + 6x2 = 0

Solving
-14 + -23x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-2.333333333 + -3.833333333x + x2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -3.833333333x + 2.333333333 + x2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -3.833333333x + x2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -3.833333333x + x2 = 0 + 2.333333333
-3.833333333x + x2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-3.833333333x + x2 = 2.333333333

The x term is -3.833333333x.  Take half its coefficient (-1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
-3.833333333x + 3.673611112 + x2 = 2.333333333 + 3.673611112

Reorder the terms:
3.673611112 + -3.833333333x + x2 = 2.333333333 + 3.673611112

Combine like terms: 2.333333333 + 3.673611112 = 6.006944445
3.673611112 + -3.833333333x + x2 = 6.006944445

Factor a perfect square on the left side:
(x + -1.916666667)(x + -1.916666667) = 6.006944445

Calculate the square root of the right side: 2.450906862

Break this problem into two subproblems by setting 
(x + -1.916666667) equal to 2.450906862 and -2.450906862.


Subproblem 1
x + -1.916666667 = 2.450906862

Simplifying
x + -1.916666667 = 2.450906862

Reorder the terms:
-1.916666667 + x = 2.450906862

Solving
-1.916666667 + x = 2.450906862

Solving for variable 'x'.

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

Add '1.916666667' to each side of the equation.
-1.916666667 + 1.916666667 + x = 2.450906862 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + x = 2.450906862 + 1.916666667
x = 2.450906862 + 1.916666667

Combine like terms: 2.450906862 + 1.916666667 = 4.367573529
x = 4.367573529

Simplifying
x = 4.367573529


Subproblem 2
x + -1.916666667 = -2.450906862

Simplifying
x + -1.916666667 = -2.450906862

Reorder the terms:
-1.916666667 + x = -2.450906862

Solving
-1.916666667 + x = -2.450906862

Solving for variable 'x'.

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

Add '1.916666667' to each side of the equation.
-1.916666667 + 1.916666667 + x = -2.450906862 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + x = -2.450906862 + 1.916666667
x = -2.450906862 + 1.916666667

Combine like terms: -2.450906862 + 1.916666667 = -0.534240195
x = -0.534240195

Simplifying
x = -0.534240195


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.367573529, -0.534240195}

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