j) \( x^{2}-3 x-2=\frac{8}{x^{2}-3 x} \)

To solve the equation \( x^{2}-3 x-2=\frac{8}{x^{2}-3 x} \), we can start by multiplying both sides by \( x^{2}-3 x \) to eliminate the fraction. This gives us: \[ (x^{2}-3 x)(x^{2}-3 x-2) = 8 \] Let \( u = x^{2} - 3x \). The equation simplifies to: \[ u(u - 2) = 8 \] This can be rewritten as a quadratic equation: \[ u^2 - 2u - 8 = 0 \] Now, we can use the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = 1 \), \( b = -2 \), and \( c = -8 \): \[ u = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)} \] Calculating the discriminant: \[ = \frac{2 \pm \sqrt{4 + 32}}{2} = \frac{2 \pm \sqrt{36}}{2} = \frac{2 \pm 6}{2} \] This gives us two potential solutions for \( u \): 1. \( u = \frac{8}{2} = 4 \) 2. \( u = \frac{-4}{2} = -2 \) Now, we’ll revert \( u \) back to \( x^{2} - 3x \): 1. For \( x^{2} - 3x = 4 \): \[ x^{2} - 3x - 4 = 0 \] Factoring gives \( (x - 4)(x + 1) = 0 \). Thus, \( x = 4 \) or \( x = -1 \). 2. For \( x^{2} - 3x = -2 \): \[ x^{2} - 3x + 2 = 0 \] Factoring gives \( (x - 2)(x - 1) = 0 \). Thus, \( x = 2 \) or \( x = 1 \). The solutions to the original equation are \( x = 4, -1, 2, 1 \).