2x^2-3x-106=0

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Solution for 2x^2-3x-106=0 equation: 


Simplifying
2x2 + -3x + -106 = 0

Reorder the terms:
-106 + -3x + 2x2 = 0

Solving
-106 + -3x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-53 + -1.5x + x2 = 0

Move the constant term to the right:

Add '53' to each side of the equation.
-53 + -1.5x + 53 + x2 = 0 + 53

Reorder the terms:
-53 + 53 + -1.5x + x2 = 0 + 53

Combine like terms: -53 + 53 = 0
0 + -1.5x + x2 = 0 + 53
-1.5x + x2 = 0 + 53

Combine like terms: 0 + 53 = 53
-1.5x + x2 = 53

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 = 53 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 53 + 0.5625

Combine like terms: 53 + 0.5625 = 53.5625
0.5625 + -1.5x + x2 = 53.5625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 53.5625

Calculate the square root of the right side: 7.318640584

Break this problem into two subproblems by setting 
(x + -0.75) equal to 7.318640584 and -7.318640584.


Subproblem 1
x + -0.75 = 7.318640584

Simplifying
x + -0.75 = 7.318640584

Reorder the terms:
-0.75 + x = 7.318640584

Solving
-0.75 + x = 7.318640584

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 = 7.318640584 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 7.318640584 + 0.75
x = 7.318640584 + 0.75

Combine like terms: 7.318640584 + 0.75 = 8.068640584
x = 8.068640584

Simplifying
x = 8.068640584


Subproblem 2
x + -0.75 = -7.318640584

Simplifying
x + -0.75 = -7.318640584

Reorder the terms:
-0.75 + x = -7.318640584

Solving
-0.75 + x = -7.318640584

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 = -7.318640584 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -7.318640584 + 0.75
x = -7.318640584 + 0.75

Combine like terms: -7.318640584 + 0.75 = -6.568640584
x = -6.568640584

Simplifying
x = -6.568640584


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.068640584, -6.568640584}

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