x+120=4x(x-120)

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Solution for x+120=4x(x-120) equation:


Simplifying
x + 120 = 4x(x + -120)

Reorder the terms:
120 + x = 4x(x + -120)

Reorder the terms:
120 + x = 4x(-120 + x)
120 + x = (-120 * 4x + x * 4x)
120 + x = (-480x + 4x2)

Solving
120 + x = -480x + 4x2

Solving for variable 'x'.

Combine like terms: x + 480x = 481x
120 + 481x + -4x2 = -480x + 4x2 + 480x + -4x2

Reorder the terms:
120 + 481x + -4x2 = -480x + 480x + 4x2 + -4x2

Combine like terms: -480x + 480x = 0
120 + 481x + -4x2 = 0 + 4x2 + -4x2
120 + 481x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
120 + 481x + -4x2 = 0

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

Divide each side by '-4'.
-30 + -120.25x + x2 = 0

Move the constant term to the right:

Add '30' to each side of the equation.
-30 + -120.25x + 30 + x2 = 0 + 30

Reorder the terms:
-30 + 30 + -120.25x + x2 = 0 + 30

Combine like terms: -30 + 30 = 0
0 + -120.25x + x2 = 0 + 30
-120.25x + x2 = 0 + 30

Combine like terms: 0 + 30 = 30
-120.25x + x2 = 30

The x term is -120.25x.  Take half its coefficient (-60.125).
Square it (3615.015625) and add it to both sides.

Add '3615.015625' to each side of the equation.
-120.25x + 3615.015625 + x2 = 30 + 3615.015625

Reorder the terms:
3615.015625 + -120.25x + x2 = 30 + 3615.015625

Combine like terms: 30 + 3615.015625 = 3645.015625
3615.015625 + -120.25x + x2 = 3645.015625

Factor a perfect square on the left side:
(x + -60.125)(x + -60.125) = 3645.015625

Calculate the square root of the right side: 60.373964794

Break this problem into two subproblems by setting 
(x + -60.125) equal to 60.373964794 and -60.373964794.

Subproblem 1

x + -60.125 = 60.373964794 Simplifying x + -60.125 = 60.373964794 Reorder the terms: -60.125 + x = 60.373964794 Solving -60.125 + x = 60.373964794 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '60.125' to each side of the equation. -60.125 + 60.125 + x = 60.373964794 + 60.125 Combine like terms: -60.125 + 60.125 = 0.000 0.000 + x = 60.373964794 + 60.125 x = 60.373964794 + 60.125 Combine like terms: 60.373964794 + 60.125 = 120.498964794 x = 120.498964794 Simplifying x = 120.498964794

Subproblem 2

x + -60.125 = -60.373964794 Simplifying x + -60.125 = -60.373964794 Reorder the terms: -60.125 + x = -60.373964794 Solving -60.125 + x = -60.373964794 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '60.125' to each side of the equation. -60.125 + 60.125 + x = -60.373964794 + 60.125 Combine like terms: -60.125 + 60.125 = 0.000 0.000 + x = -60.373964794 + 60.125 x = -60.373964794 + 60.125 Combine like terms: -60.373964794 + 60.125 = -0.248964794 x = -0.248964794 Simplifying x = -0.248964794

Solution

The solution to the problem is based on the solutions from the subproblems. x = {120.498964794, -0.248964794}

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