2x^2-31x-105=0

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Solution for 2x^2-31x-105=0 equation: 


Simplifying
2x2 + -31x + -105 = 0

Reorder the terms:
-105 + -31x + 2x2 = 0

Solving
-105 + -31x + 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'.
-52.5 + -15.5x + x2 = 0

Move the constant term to the right:

Add '52.5' to each side of the equation.
-52.5 + -15.5x + 52.5 + x2 = 0 + 52.5

Reorder the terms:
-52.5 + 52.5 + -15.5x + x2 = 0 + 52.5

Combine like terms: -52.5 + 52.5 = 0.0
0.0 + -15.5x + x2 = 0 + 52.5
-15.5x + x2 = 0 + 52.5

Combine like terms: 0 + 52.5 = 52.5
-15.5x + x2 = 52.5

The x term is -15.5x.  Take half its coefficient (-7.75).
Square it (60.0625) and add it to both sides.

Add '60.0625' to each side of the equation.
-15.5x + 60.0625 + x2 = 52.5 + 60.0625

Reorder the terms:
60.0625 + -15.5x + x2 = 52.5 + 60.0625

Combine like terms: 52.5 + 60.0625 = 112.5625
60.0625 + -15.5x + x2 = 112.5625

Factor a perfect square on the left side:
(x + -7.75)(x + -7.75) = 112.5625

Calculate the square root of the right side: 10.609547587

Break this problem into two subproblems by setting 
(x + -7.75) equal to 10.609547587 and -10.609547587.


Subproblem 1
x + -7.75 = 10.609547587

Simplifying
x + -7.75 = 10.609547587

Reorder the terms:
-7.75 + x = 10.609547587

Solving
-7.75 + x = 10.609547587

Solving for variable 'x'.

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

Add '7.75' to each side of the equation.
-7.75 + 7.75 + x = 10.609547587 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + x = 10.609547587 + 7.75
x = 10.609547587 + 7.75

Combine like terms: 10.609547587 + 7.75 = 18.359547587
x = 18.359547587

Simplifying
x = 18.359547587


Subproblem 2
x + -7.75 = -10.609547587

Simplifying
x + -7.75 = -10.609547587

Reorder the terms:
-7.75 + x = -10.609547587

Solving
-7.75 + x = -10.609547587

Solving for variable 'x'.

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

Add '7.75' to each side of the equation.
-7.75 + 7.75 + x = -10.609547587 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + x = -10.609547587 + 7.75
x = -10.609547587 + 7.75

Combine like terms: -10.609547587 + 7.75 = -2.859547587
x = -2.859547587

Simplifying
x = -2.859547587


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {18.359547587, -2.859547587}

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