2) \( \frac{x^{2}-4}{x-2}=0 \)

To solve the equation \( \frac{x^{2}-4}{x-2}=0 \), we need to recognize that a fraction is equal to zero when its numerator is zero (as long as the denominator isn’t zero). So, we set the numerator \( x^{2} - 4 = 0 \). Factoring this, we get \( (x-2)(x+2) = 0 \). Therefore, \( x-2 = 0 \) or \( x+2 = 0 \), leading to the solutions \( x = 2 \) or \( x = -2 \). However, we need to note that substituting \( x = 2 \) makes the denominator zero, thus it is an extraneous solution. Hence, our only valid solution is \( x = -2 \). Now, onto some practical insights! Did you know that this kind of equation is commonly applied in real-world scenarios, such as determining the points where a quantity reaches zero? Whether it’s assessing profits that will hit break-even points or figuring out where a chemical concentration becomes negligible, algebra helps make the magic happen! Missteps can often occur when dealing with rational expressions. A classic blunder is ignoring restrictions caused by the denominator. Always check if substituting your solutions back into the original equation results in any undefined values! In this example, \( x = 2 \) is a perfect demonstration of this pitfall, so be wary and double-check your results!