Solving $5z^2 - 85 = 0$ Using The Square Root Property

Hey guys! Today, let's dive into the fascinating world of quadratic equations and how to solve them using a neat trick called the square root property. This method is super handy when our equation is in a specific form, making it much easier to find the solutions. We'll break down the concept, walk through an example, and highlight why this method is so valuable. So, let's jump right in!

Understanding the Square Root Property

The square root property is a powerful tool for solving quadratic equations, especially those that can be written in the form ax² + c = 0. In simpler terms, it’s most effective when you have a squared term on one side of the equation and a constant on the other. The core idea behind this property is that if x² = k, then x = √k or x = -√k. This is because both the positive and negative square roots of k, when squared, will give you k. Let's think about it with an example. Say you have x² = 9. What numbers, when squared, give you 9? Well, both 3 and -3 do! That's why we need to consider both the positive and negative roots.

The real magic of the square root property lies in its simplicity and efficiency. When you encounter a quadratic equation in the ax² + c = 0 format, using this property can save you a lot of time and effort compared to other methods like factoring or using the quadratic formula. The property allows us to isolate the squared variable term and then directly take the square root of both sides, immediately revealing the possible solutions. It's like having a shortcut that cuts through the complexity and gets you straight to the answer. But, it's not just about speed; it's also about understanding the nature of quadratic equations and their solutions. By using the square root property, we're visually and conceptually reinforcing the idea that quadratic equations often have two solutions, reflecting the symmetry inherent in the squaring operation. Remember, though, this method shines brightest when the equation is in the correct form. If your equation has an x term (like ax² + bx + c = 0), you'll need to reach for other techniques, like factoring, completing the square, or the trusty quadratic formula. The square root property is a specialized tool in our mathematical toolkit, perfect for the right job.

Solving the Equation: 5z² - 85 = 0

Let's tackle the equation 5z² - 85 = 0 using the square root property. The first step is to isolate the squared term, which in this case is z². We want to get the equation into a form where we have z² alone on one side. To do this, we'll start by adding 85 to both sides of the equation. This gives us 5z² = 85. We're one step closer to isolating z², but we still have that pesky 5 multiplying it. No problem! We can easily get rid of it by dividing both sides of the equation by 5. This leaves us with z² = 17. Now we're talking! We have the equation in the perfect form to apply the square root property.

Now that we have z² = 17, we can take the square root of both sides. Remember, when we do this, we need to consider both the positive and negative square roots. This is because both √17 squared and -√17 squared will give us 17. So, we get z = √17 or z = -√17. These are our two solutions! It’s important to express both the positive and negative roots to ensure we capture all possible solutions to the quadratic equation. You might be wondering,