To begin with, we need to simplify the given expression as follows:

X² - (x + 2) (x - 3) / (7x - 1) = 2/3

X² - (x² - x - 6) / (7x - 1) = 2/3

Multiplying both sides by (7x - 1):

(7x - 1) X² - (x² - x - 6) = (2/3) (7x - 1)

7x³ - x² - 6x - 2 = 14x/3 - 2/3

Multiplying both sides by 3:

21x³ - 3x² - 18x - 6 = 14x - 2

21x³ - 3x² - 32x + 4 = 0


Now we need to factor the equation to find the values of x that satisfy it.

21x³ - 3x² - 32x + 4 = 0

3(7x² - x - 2) (x - 2) = 0

We can solve for x by using the Zero Product Property:

7x² - x - 2 = 0 or x - 2 = 0

Solving the first equation using quadratic formula, we get:

x = (1 ± √(1 + 4(7)(2))) / (2(7))

x = (1 ± √(57)) / 14

Therefore, the solutions to the equation are:

x = (1 + √(57)) / 14 or x = (1 - √(57)) / 14 or x = 2

However, we need to check if these values make the denominator zero. If they do, they are not valid solutions.


Checking for x = (1 + √(57)) / 14:

7x - 1 = 7(1 + √(57)) / 14 - 1 = (√(57) + 6) / 2

Since this is not equal to zero, x = (1 + √(57)) / 14 is a valid solution.

Checking for x = (1 - √(57)) / 14:

7x - 1 = 7(1 - √(57)) / 14 - 1 = (-√(57) + 6) / 2

Since this is not equal to zero, x = (1 - √(57)) / 14 is also a valid solution.

Checking for x = 2:

7x - 1 = 7(2) - 1 = 13

Since this is not equal to zero, x = 2 is also a valid solution.

Therefore, the solutions to the equation are:

x = (1 + √(57)) / 14, x = (1 - √(57)) / 14, and x = 2.

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