k) \( 6 x\left(x^{2}-36\right)=0 \)

To solve the equation \( 6x(x^{2}-36)=0 \), we can break it down into two parts: \( 6x = 0 \) and \( x^{2} - 36 = 0 \). From \( 6x = 0 \), we find that \( x = 0 \). For the second part, \( x^{2} - 36 = 0 \) factors into \( (x-6)(x+6) = 0 \), giving us the solutions \( x = 6 \) and \( x = -6 \). Therefore, the complete solution set is \( x = 0, 6, -6 \). Now, let’s consider contexts where this kind of equation can pop up! Quadratic equations like this one often show up in physics, particularly in projectile motion problems or anywhere forces are acting in opposing directions. The zeroes can represent points of no net movement or changes in direction, making them crucial for understanding motion in a world governed by physics. And if you want to avoid those pesky mistakes while solving similar equations, remember to keep track of your factors and don’t forget about the zero-product property! Always set each factor to zero separately; it's a common pitfall to overlook one of the solutions. Double-checking each step can save you from an embarrassing miscalculation!