45(x-1)60(x-2)=60(x-2)-15x

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Solution for 45(x-1)60(x-2)=60(x-2)-15x equation: 


Simplifying
45(x + -1) * 60(x + -2) = 60(x + -2) + -15x

Reorder the terms:
45(-1 + x) * 60(x + -2) = 60(x + -2) + -15x

Reorder the terms:
45(-1 + x) * 60(-2 + x) = 60(x + -2) + -15x

Reorder the terms for easier multiplication:
45 * 60(-1 + x)(-2 + x) = 60(x + -2) + -15x

Multiply 45 * 60
2700(-1 + x)(-2 + x) = 60(x + -2) + -15x

Multiply (-1 + x) * (-2 + x)
2700(-1(-2 + x) + x(-2 + x)) = 60(x + -2) + -15x
2700((-2 * -1 + x * -1) + x(-2 + x)) = 60(x + -2) + -15x
2700((2 + -1x) + x(-2 + x)) = 60(x + -2) + -15x
2700(2 + -1x + (-2 * x + x * x)) = 60(x + -2) + -15x
2700(2 + -1x + (-2x + x2)) = 60(x + -2) + -15x

Combine like terms: -1x + -2x = -3x
2700(2 + -3x + x2) = 60(x + -2) + -15x
(2 * 2700 + -3x * 2700 + x2 * 2700) = 60(x + -2) + -15x
(5400 + -8100x + 2700x2) = 60(x + -2) + -15x

Reorder the terms:
5400 + -8100x + 2700x2 = 60(-2 + x) + -15x
5400 + -8100x + 2700x2 = (-2 * 60 + x * 60) + -15x
5400 + -8100x + 2700x2 = (-120 + 60x) + -15x

Combine like terms: 60x + -15x = 45x
5400 + -8100x + 2700x2 = -120 + 45x

Solving
5400 + -8100x + 2700x2 = -120 + 45x

Solving for variable 'x'.

Reorder the terms:
5400 + 120 + -8100x + -45x + 2700x2 = -120 + 45x + 120 + -45x

Combine like terms: 5400 + 120 = 5520
5520 + -8100x + -45x + 2700x2 = -120 + 45x + 120 + -45x

Combine like terms: -8100x + -45x = -8145x
5520 + -8145x + 2700x2 = -120 + 45x + 120 + -45x

Reorder the terms:
5520 + -8145x + 2700x2 = -120 + 120 + 45x + -45x

Combine like terms: -120 + 120 = 0
5520 + -8145x + 2700x2 = 0 + 45x + -45x
5520 + -8145x + 2700x2 = 45x + -45x

Combine like terms: 45x + -45x = 0
5520 + -8145x + 2700x2 = 0

Factor out the Greatest Common Factor (GCF), '15'.
15(368 + -543x + 180x2) = 0

Ignore the factor 15.

Subproblem 1
Set the factor '(368 + -543x + 180x2)' equal to zero and attempt to solve:

Simplifying
368 + -543x + 180x2 = 0

Solving
368 + -543x + 180x2 = 0

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

Divide each side by '180'.
2.044444444 + -3.016666667x + x2 = 0.0

Move the constant term to the right:

Add '-2.044444444' to each side of the equation.
2.044444444 + -3.016666667x + -2.044444444 + x2 = 0.0 + -2.044444444

Reorder the terms:
2.044444444 + -2.044444444 + -3.016666667x + x2 = 0.0 + -2.044444444

Combine like terms: 2.044444444 + -2.044444444 = 0.000000000
0.000000000 + -3.016666667x + x2 = 0.0 + -2.044444444
-3.016666667x + x2 = 0.0 + -2.044444444

Combine like terms: 0.0 + -2.044444444 = -2.044444444
-3.016666667x + x2 = -2.044444444

The x term is -3.016666667x.  Take half its coefficient (-1.508333334).
Square it (2.275069446) and add it to both sides.

Add '2.275069446' to each side of the equation.
-3.016666667x + 2.275069446 + x2 = -2.044444444 + 2.275069446

Reorder the terms:
2.275069446 + -3.016666667x + x2 = -2.044444444 + 2.275069446

Combine like terms: -2.044444444 + 2.275069446 = 0.230625002
2.275069446 + -3.016666667x + x2 = 0.230625002

Factor a perfect square on the left side:
(x + -1.508333334)(x + -1.508333334) = 0.230625002

Calculate the square root of the right side: 0.48023432

Break this problem into two subproblems by setting 
(x + -1.508333334) equal to 0.48023432 and -0.48023432.


Subproblem 1
x + -1.508333334 = 0.48023432

Simplifying
x + -1.508333334 = 0.48023432

Reorder the terms:
-1.508333334 + x = 0.48023432

Solving
-1.508333334 + x = 0.48023432

Solving for variable 'x'.

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

Add '1.508333334' to each side of the equation.
-1.508333334 + 1.508333334 + x = 0.48023432 + 1.508333334

Combine like terms: -1.508333334 + 1.508333334 = 0.000000000
0.000000000 + x = 0.48023432 + 1.508333334
x = 0.48023432 + 1.508333334

Combine like terms: 0.48023432 + 1.508333334 = 1.988567654
x = 1.988567654

Simplifying
x = 1.988567654


Subproblem 2
x + -1.508333334 = -0.48023432

Simplifying
x + -1.508333334 = -0.48023432

Reorder the terms:
-1.508333334 + x = -0.48023432

Solving
-1.508333334 + x = -0.48023432

Solving for variable 'x'.

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

Add '1.508333334' to each side of the equation.
-1.508333334 + 1.508333334 + x = -0.48023432 + 1.508333334

Combine like terms: -1.508333334 + 1.508333334 = 0.000000000
0.000000000 + x = -0.48023432 + 1.508333334
x = -0.48023432 + 1.508333334

Combine like terms: -0.48023432 + 1.508333334 = 1.028099014
x = 1.028099014

Simplifying
x = 1.028099014


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.988567654, 1.028099014}
Solution
x = {1.988567654, 1.028099014}

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