6x^2+x-75=0

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Solution for 6x^2+x-75=0 equation: 


Simplifying
6x2 + x + -75 = 0

Reorder the terms:
-75 + x + 6x2 = 0

Solving
-75 + x + 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'.
-12.5 + 0.1666666667x + x2 = 0

Move the constant term to the right:

Add '12.5' to each side of the equation.
-12.5 + 0.1666666667x + 12.5 + x2 = 0 + 12.5

Reorder the terms:
-12.5 + 12.5 + 0.1666666667x + x2 = 0 + 12.5

Combine like terms: -12.5 + 12.5 = 0.0
0.0 + 0.1666666667x + x2 = 0 + 12.5
0.1666666667x + x2 = 0 + 12.5

Combine like terms: 0 + 12.5 = 12.5
0.1666666667x + x2 = 12.5

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1666666667x + 0.25 + x2 = 12.5 + 0.25

Reorder the terms:
0.25 + 0.1666666667x + x2 = 12.5 + 0.25

Combine like terms: 12.5 + 0.25 = 12.75
0.25 + 0.1666666667x + x2 = 12.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 12.75

Calculate the square root of the right side: 3.570714214

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.570714214 and -3.570714214.


Subproblem 1
x + 0.5 = 3.570714214

Simplifying
x + 0.5 = 3.570714214

Reorder the terms:
0.5 + x = 3.570714214

Solving
0.5 + x = 3.570714214

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 3.570714214 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 3.570714214 + -0.5
x = 3.570714214 + -0.5

Combine like terms: 3.570714214 + -0.5 = 3.070714214
x = 3.070714214

Simplifying
x = 3.070714214


Subproblem 2
x + 0.5 = -3.570714214

Simplifying
x + 0.5 = -3.570714214

Reorder the terms:
0.5 + x = -3.570714214

Solving
0.5 + x = -3.570714214

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -3.570714214 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -3.570714214 + -0.5
x = -3.570714214 + -0.5

Combine like terms: -3.570714214 + -0.5 = -4.070714214
x = -4.070714214

Simplifying
x = -4.070714214


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.070714214, -4.070714214}

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