(2x-1)(3x-5)+x=180

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Solution for (2x-1)(3x-5)+x=180 equation:


Simplifying
(2x + -1)(3x + -5) + x = 180

Reorder the terms:
(-1 + 2x)(3x + -5) + x = 180

Reorder the terms:
(-1 + 2x)(-5 + 3x) + x = 180

Multiply (-1 + 2x) * (-5 + 3x)
(-1(-5 + 3x) + 2x * (-5 + 3x)) + x = 180
((-5 * -1 + 3x * -1) + 2x * (-5 + 3x)) + x = 180
((5 + -3x) + 2x * (-5 + 3x)) + x = 180
(5 + -3x + (-5 * 2x + 3x * 2x)) + x = 180
(5 + -3x + (-10x + 6x2)) + x = 180

Combine like terms: -3x + -10x = -13x
(5 + -13x + 6x2) + x = 180

Reorder the terms:
5 + -13x + x + 6x2 = 180

Combine like terms: -13x + x = -12x
5 + -12x + 6x2 = 180

Solving
5 + -12x + 6x2 = 180

Solving for variable 'x'.

Reorder the terms:
5 + -180 + -12x + 6x2 = 180 + -180

Combine like terms: 5 + -180 = -175
-175 + -12x + 6x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-175 + -12x + 6x2 = 0

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

Divide each side by '6'.
-29.16666667 + -2x + x2 = 0

Move the constant term to the right:

Add '29.16666667' to each side of the equation.
-29.16666667 + -2x + 29.16666667 + x2 = 0 + 29.16666667

Reorder the terms:
-29.16666667 + 29.16666667 + -2x + x2 = 0 + 29.16666667

Combine like terms: -29.16666667 + 29.16666667 = 0.00000000
0.00000000 + -2x + x2 = 0 + 29.16666667
-2x + x2 = 0 + 29.16666667

Combine like terms: 0 + 29.16666667 = 29.16666667
-2x + x2 = 29.16666667

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 29.16666667 + 1

Reorder the terms:
1 + -2x + x2 = 29.16666667 + 1

Combine like terms: 29.16666667 + 1 = 30.16666667
1 + -2x + x2 = 30.16666667

Factor a perfect square on the left side:
(x + -1)(x + -1) = 30.16666667

Calculate the square root of the right side: 5.492419018

Break this problem into two subproblems by setting 
(x + -1) equal to 5.492419018 and -5.492419018.

Subproblem 1

x + -1 = 5.492419018 Simplifying x + -1 = 5.492419018 Reorder the terms: -1 + x = 5.492419018 Solving -1 + x = 5.492419018 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 5.492419018 + 1 Combine like terms: -1 + 1 = 0 0 + x = 5.492419018 + 1 x = 5.492419018 + 1 Combine like terms: 5.492419018 + 1 = 6.492419018 x = 6.492419018 Simplifying x = 6.492419018

Subproblem 2

x + -1 = -5.492419018 Simplifying x + -1 = -5.492419018 Reorder the terms: -1 + x = -5.492419018 Solving -1 + x = -5.492419018 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -5.492419018 + 1 Combine like terms: -1 + 1 = 0 0 + x = -5.492419018 + 1 x = -5.492419018 + 1 Combine like terms: -5.492419018 + 1 = -4.492419018 x = -4.492419018 Simplifying x = -4.492419018

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.492419018, -4.492419018}

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