4x^2+x-120=0

Solution for 4x^2+x-120=0 equation:

Simplifying
4x2 + x + -120 = 0

Reorder the terms:
-120 + x + 4x2 = 0

Solving
-120 + x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-30 + 0.25x + x2 = 0

Move the constant term to the right:

Add '30' to each side of the equation.
-30 + 0.25x + 30 + x2 = 0 + 30

Reorder the terms:
-30 + 30 + 0.25x + x2 = 0 + 30

Combine like terms: -30 + 30 = 0
0 + 0.25x + x2 = 0 + 30
0.25x + x2 = 0 + 30

Combine like terms: 0 + 30 = 30
0.25x + x2 = 30

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.
0.25x + 0.25 + x2 = 30 + 0.25

Reorder the terms:
0.25 + 0.25x + x2 = 30 + 0.25

Combine like terms: 30 + 0.25 = 30.25
0.25 + 0.25x + x2 = 30.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 30.25

Calculate the square root of the right side: 5.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 5.5 and -5.5.

Subproblem 1
x + 0.5 = 5.5

Simplifying
x + 0.5 = 5.5

Reorder the terms:
0.5 + x = 5.5

Solving
0.5 + x = 5.5

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 = 5.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 5.5 + -0.5
x = 5.5 + -0.5

Combine like terms: 5.5 + -0.5 = 5
x = 5

Simplifying
x = 5

Subproblem 2
x + 0.5 = -5.5

Simplifying
x + 0.5 = -5.5

Reorder the terms:
0.5 + x = -5.5

Solving
0.5 + x = -5.5

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 = -5.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -5.5 + -0.5
x = -5.5 + -0.5

Combine like terms: -5.5 + -0.5 = -6
x = -6

Simplifying
x = -6

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5, -6}