Roots Of Quadratic Function F(b) = B^2 - 75

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Hey guys! Let's dive into finding the roots of the quadratic function f(b) = b^2 - 75. This is a classic math problem, and we're going to break it down step by step. We need to identify which two options from the given choices are the correct roots. So, let’s put on our math hats and get started!

Understanding Quadratic Roots

First, let's get the basics straight. The roots of a quadratic function are the values of b for which f(b) = 0. In simpler terms, these are the points where the graph of the function crosses the x-axis. For a quadratic function in the form f(b) = ab^2 + cb + d, the roots can be found by setting the function equal to zero and solving for b. This often involves factoring, completing the square, or using the quadratic formula. But in our case, we have a simpler form, which makes it easier to solve. We'll see how in just a bit!

For our specific function, f(b) = b^2 - 75, we need to find the values of b that make this equation true when it equals zero. This means we are looking for b such that b^2 - 75 = 0. Understanding this basic principle is key to solving not just this problem, but many other quadratic equations you might encounter. Remember, the roots tell us a lot about the function’s behavior and graph, so finding them is a crucial skill in algebra.

Solving for the Roots

Now, let’s roll up our sleeves and find the roots of f(b) = b^2 - 75. To do this, we set the function equal to zero:

b^2 - 75 = 0

Our goal is to isolate b. The first step is to add 75 to both sides of the equation:

b^2 = 75

Now, we need to get b by itself. Since b is squared, we’ll take the square root of both sides. Remember, when we take the square root, we need to consider both the positive and negative roots, because both a positive and a negative number, when squared, will give a positive result. So we get:

b = ±√75

But we’re not quite done yet! We need to simplify the square root of 75. Think of 75 as a product of its factors. We can write 75 as 25 times 3, where 25 is a perfect square. This helps us simplify the square root:

b = ±√(25 * 3)

We can break this down further using the property of square roots that says √(a * c) = √a * √c:

b = ±√25 * √3

Since the square root of 25 is 5, we have:

b = ±5√3

So, the roots of the quadratic function f(b) = b^2 - 75 are b = 5√3 and b = -5√3. These are the two values of b that make the function equal to zero.

Matching the Solutions

Alright, we've found the roots: b = 5√3 and b = -5√3. Now, let's see which options from our list match these solutions. Looking at the options provided:

A. b = 5√3 B. b = -5√3 C. b = 3√5 D. b = -3√5 E. b = 25√3

It’s pretty clear that options A and B match our solutions perfectly. Option A, b = 5√3, is one of our roots, and option B, b = -5√3, is the other. Options C and D have the numbers 3 and 5 swapped inside the square root, so they are not correct. And option E is way off with 25√3. Therefore, the correct answers are options A and B.

This exercise highlights the importance of not only solving the equation correctly but also carefully comparing your solutions with the given options. It’s easy to make a mistake if you rush this final step!

Why These are the Roots

Let's take a moment to really understand why b = 5√3 and b = -5√3 are the roots of the function f(b) = b^2 - 75. Remember, roots are the values of b that make f(b) = 0. We can verify our solutions by plugging them back into the original equation and seeing if they make the equation true.

First, let's try b = 5√3:

f(5√3) = (5√3)^2 - 75

To square 5√3, we square both the 5 and the √3:

(5√3)^2 = 5^2 * (√3)^2 = 25 * 3 = 75

So, our equation becomes:

f(5√3) = 75 - 75 = 0

Great! It works. Now let's try b = -5√3:

f(-5√3) = (-5√3)^2 - 75

Again, we square both the -5 and the √3:

(-5√3)^2 = (-5)^2 * (√3)^2 = 25 * 3 = 75

So, our equation becomes:

f(-5√3) = 75 - 75 = 0

Excellent! This one works too. This verification step is super important because it gives us confidence that our solutions are correct. It’s a good habit to get into, especially on exams!

The roots 5√3 and -5√3 tell us where the parabola f(b) = b^2 - 75 intersects the x-axis. Since one root is the positive version and the other is the negative version, the parabola is symmetric about the y-axis, and its vertex is at (0, -75). Understanding these connections between the roots and the graph can give you a deeper insight into quadratic functions.

Tips for Solving Similar Problems

Solving for roots of quadratic functions is a fundamental skill in algebra, and there are a few tips and tricks that can make the process smoother. Let’s go over some helpful strategies for tackling problems like this one.

  1. Simplify First: Whenever you're dealing with equations, always look for opportunities to simplify. In our case, we had a simple quadratic equation where isolating b^2 was straightforward. Simplifying the equation before diving into more complex methods can save you time and reduce the chance of errors.

  2. Consider Both Positive and Negative Roots: When taking the square root of both sides of an equation, remember to include both the positive and negative roots. This is a common mistake, and forgetting the negative root can lead to an incomplete solution. In our problem, we had b^2 = 75, and we needed to remember that both √75 and -√75 are valid solutions.

  3. Simplify Radicals: Always simplify radicals as much as possible. This often involves factoring the number under the square root and looking for perfect square factors. We simplified √75 by recognizing that 75 is 25 times 3, and 25 is a perfect square. Simplifying radicals not only gives you the most accurate answer but also makes it easier to compare your answer with the options provided.

  4. Check Your Answers: As we did earlier, plug your solutions back into the original equation to verify that they are correct. This is a crucial step in catching any mistakes and ensuring that your solutions are valid. It's a bit like double-checking your work in any important task.

  5. Understand the Concept of Roots: Remember that the roots of a quadratic function are the points where the function equals zero. These are also the x-intercepts of the graph of the function. Having a clear understanding of what roots represent can help you visualize the problem and ensure your solutions make sense.

  6. Practice Regularly: Like any math skill, solving quadratic equations becomes easier with practice. Work through a variety of problems, and don't be afraid to try different methods. The more you practice, the more comfortable and confident you'll become.

By keeping these tips in mind, you’ll be well-equipped to solve similar problems and master the art of finding roots of quadratic functions. Math can be challenging, but with the right strategies and a bit of practice, you can conquer any problem!

Conclusion

So, to wrap things up, the roots of the quadratic function f(b) = b^2 - 75 are indeed b = 5√3 and b = -5√3, which correspond to options A and B. We walked through the process of setting the function to zero, isolating b, simplifying the square root, and verifying our answers. Remember, solving for roots is a key skill in algebra, and by mastering these techniques, you'll be well-prepared for more advanced topics. Keep practicing, and you'll become a pro at solving quadratic equations in no time! You got this!