x(5x-2)(7x-3)=x

Solution for x(5x-2)(7x-3)=x equation:

Simplifying
x(5x + -2)(7x + -3) = x

Reorder the terms:
x(-2 + 5x)(7x + -3) = x

Reorder the terms:
x(-2 + 5x)(-3 + 7x) = x

Multiply (-2 + 5x) * (-3 + 7x)
x(-2(-3 + 7x) + 5x * (-3 + 7x)) = x
x((-3 * -2 + 7x * -2) + 5x * (-3 + 7x)) = x
x((6 + -14x) + 5x * (-3 + 7x)) = x
x(6 + -14x + (-3 * 5x + 7x * 5x)) = x
x(6 + -14x + (-15x + 35x2)) = x

Combine like terms: -14x + -15x = -29x
x(6 + -29x + 35x2) = x
(6 * x + -29x * x + 35x2 * x) = x
(6x + -29x2 + 35x3) = x

Solving
6x + -29x2 + 35x3 = x

Solving for variable 'x'.

Reorder the terms:
6x + -1x + -29x2 + 35x3 = x + -1x

Combine like terms: 6x + -1x = 5x
5x + -29x2 + 35x3 = x + -1x

Combine like terms: x + -1x = 0
5x + -29x2 + 35x3 = 0

Factor out the Greatest Common Factor (GCF), 'x'.
x(5 + -29x + 35x2) = 0

Subproblem 1
Set the factor 'x' equal to zero and attempt to solve:

Simplifying
x = 0

Solving
x = 0

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

Simplifying
x = 0
Subproblem 2
Set the factor '(5 + -29x + 35x2)' equal to zero and attempt to solve:

Simplifying
5 + -29x + 35x2 = 0

Solving
5 + -29x + 35x2 = 0

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

Divide each side by '35'.
0.1428571429 + -0.8285714286x + x2 = 0

Move the constant term to the right:

Add '-0.1428571429' to each side of the equation.
0.1428571429 + -0.8285714286x + -0.1428571429 + x2 = 0 + -0.1428571429

Reorder the terms:
0.1428571429 + -0.1428571429 + -0.8285714286x + x2 = 0 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + -0.8285714286x + x2 = 0 + -0.1428571429
-0.8285714286x + x2 = 0 + -0.1428571429

Combine like terms: 0 + -0.1428571429 = -0.1428571429
-0.8285714286x + x2 = -0.1428571429

The x term is -0.8285714286x.  Take half its coefficient (-0.4142857143).
Square it (0.1716326531) and add it to both sides.

Add '0.1716326531' to each side of the equation.
-0.8285714286x + 0.1716326531 + x2 = -0.1428571429 + 0.1716326531

Reorder the terms:
0.1716326531 + -0.8285714286x + x2 = -0.1428571429 + 0.1716326531

Combine like terms: -0.1428571429 + 0.1716326531 = 0.0287755102
0.1716326531 + -0.8285714286x + x2 = 0.0287755102

Factor a perfect square on the left side:
(x + -0.4142857143)(x + -0.4142857143) = 0.0287755102

Calculate the square root of the right side: 0.169633458

Break this problem into two subproblems by setting 
(x + -0.4142857143) equal to 0.169633458 and -0.169633458.

Subproblem 1
x + -0.4142857143 = 0.169633458

Simplifying
x + -0.4142857143 = 0.169633458

Reorder the terms:
-0.4142857143 + x = 0.169633458

Solving
-0.4142857143 + x = 0.169633458

Solving for variable 'x'.

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

Add '0.4142857143' to each side of the equation.
-0.4142857143 + 0.4142857143 + x = 0.169633458 + 0.4142857143

Combine like terms: -0.4142857143 + 0.4142857143 = 0.0000000000
0.0000000000 + x = 0.169633458 + 0.4142857143
x = 0.169633458 + 0.4142857143

Combine like terms: 0.169633458 + 0.4142857143 = 0.5839191723
x = 0.5839191723

Simplifying
x = 0.5839191723

Subproblem 2
x + -0.4142857143 = -0.169633458

Simplifying
x + -0.4142857143 = -0.169633458

Reorder the terms:
-0.4142857143 + x = -0.169633458

Solving
-0.4142857143 + x = -0.169633458

Solving for variable 'x'.

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

Add '0.4142857143' to each side of the equation.
-0.4142857143 + 0.4142857143 + x = -0.169633458 + 0.4142857143

Combine like terms: -0.4142857143 + 0.4142857143 = 0.0000000000
0.0000000000 + x = -0.169633458 + 0.4142857143
x = -0.169633458 + 0.4142857143

Combine like terms: -0.169633458 + 0.4142857143 = 0.2446522563
x = 0.2446522563

Simplifying
x = 0.2446522563

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.5839191723, 0.2446522563}
Solution
x = {0, 0.5839191723, 0.2446522563}