3. \( x^{2}-16 x+64=4 \)

To solve the equation \( x^{2}-16 x+64=4 \), first, we can simplify it by bringing all terms to one side. This leads us to \( x^{2}-16 x+60=0 \). Next, we can factor the quadratic equation. It factors to \( (x-10)(x-6)=0 \), giving us the solutions \( x=10 \) and \( x=6 \). Finally, checking these values, we find they satisfy the original equation. So, \( x=10 \) and \( x=6 \) are our answers! Also, did you know that the process of factorization in algebra wasn't really standardized until much later in history? Greeks like Euclid contributed to the understanding of numerical relationships, laying the groundwork for the algebraic techniques we use today! In real-world applications, quadratic equations like this often show up in various fields like physics or finance, for instance, when calculating the optimal price to maximize profit or determining the trajectory of a basketball! They’re not just numbers; they can represent something tangible and useful in everyday life!