e) \( w^{2}-2 w-15= \)

To solve the equation \( w^{2}-2w-15=0 \), you'll want to factor the quadratic. Look for two numbers that multiply to -15 and add up to -2. These numbers are -5 and 3. So, we can factor the equation as \( (w-5)(w+3)=0 \). Setting each factor to zero gives the solutions: \( w = 5 \) and \( w = -3 \). Now that you've got those solutions, let’s check how they can be relevant! You can use these kinds of quadratic equations for various practical situations, such as determining the maximum height of a projectile, optimizing a given area, or even financial calculations like profit maximization. Who knew math could help you become a backyard physicist or a budding entrepreneur?! Make sure to always double-check your factoring by plugging the numbers back into the original equation. A common mistake is accidentally multiplying or adding the numbers wrong. Writing out the full quadratic formula is another route if you're stuck with factoring, just in case your numbers are trickier to spot!