6x^2-9x-14=0

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Solution for 6x^2-9x-14=0 equation: 


Simplifying
6x2 + -9x + -14 = 0

Reorder the terms:
-14 + -9x + 6x2 = 0

Solving
-14 + -9x + 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 + -1.5x + x2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -1.5x + 2.333333333 + x2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -1.5x + x2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -1.5x + x2 = 0 + 2.333333333
-1.5x + x2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-1.5x + x2 = 2.333333333

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 = 2.333333333 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 2.333333333 + 0.5625

Combine like terms: 2.333333333 + 0.5625 = 2.895833333
0.5625 + -1.5x + x2 = 2.895833333

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 2.895833333

Calculate the square root of the right side: 1.701714821

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.701714821 and -1.701714821.


Subproblem 1
x + -0.75 = 1.701714821

Simplifying
x + -0.75 = 1.701714821

Reorder the terms:
-0.75 + x = 1.701714821

Solving
-0.75 + x = 1.701714821

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 = 1.701714821 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 1.701714821 + 0.75
x = 1.701714821 + 0.75

Combine like terms: 1.701714821 + 0.75 = 2.451714821
x = 2.451714821

Simplifying
x = 2.451714821


Subproblem 2
x + -0.75 = -1.701714821

Simplifying
x + -0.75 = -1.701714821

Reorder the terms:
-0.75 + x = -1.701714821

Solving
-0.75 + x = -1.701714821

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 = -1.701714821 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -1.701714821 + 0.75
x = -1.701714821 + 0.75

Combine like terms: -1.701714821 + 0.75 = -0.951714821
x = -0.951714821

Simplifying
x = -0.951714821


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.451714821, -0.951714821}

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