Finding Roots: Solve H(x) = 8x + 10. What Is The Zero?

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Finding the Roots of a Function: A Step-by-Step Guide to Solving h(x) = 8x + 10

Hey guys! Today, we're diving into the exciting world of functions and their roots. Specifically, we're going to tackle the function h(x) = 8x + 10 and figure out its zeros (or roots). This might sound intimidating, but trust me, it's totally doable! We'll break it down step by step so you can master this skill. So, let's get started and unravel the mystery of finding roots!

Understanding Roots and Zeros

First things first, what exactly are we looking for when we talk about the "roots" or "zeros" of a function? Simply put, the roots of a function are the values of x that make the function equal to zero. Graphically, these are the points where the function's graph intersects the x-axis. Think of it like this: we're searching for the x-values that, when plugged into our function, result in an output of zero.

Now, why is this important? Well, finding roots is a fundamental concept in algebra and calculus, and it has applications in various fields like physics, engineering, and economics. It helps us understand the behavior of functions and solve equations. For example, in a real-world scenario, the roots might represent the points where a projectile hits the ground or the equilibrium points in a system. So, understanding how to find them is a crucial skill to have.

In the case of our function, h(x) = 8x + 10, we want to find the value (or values) of x that make h(x) equal to zero. This means we need to solve the equation 8x + 10 = 0. Don't worry if that sounds a bit confusing right now; we're about to break down the process step-by-step.

Solving for the Roots of h(x) = 8x + 10

Okay, let's get down to business and solve for the roots of our function. Remember, our goal is to find the value of x that makes h(x) = 8x + 10 equal to zero. Here's how we do it:

  1. Set the function equal to zero: This is our first and most crucial step. We're essentially translating our problem into an equation we can solve. So, we write: 8x + 10 = 0

  2. Isolate the term with x: To do this, we need to get rid of the constant term (+10) on the left side of the equation. We can do this by subtracting 10 from both sides of the equation. This keeps the equation balanced and allows us to isolate the x term: 8x + 10 - 10 = 0 - 10. This simplifies to 8x = -10.

  3. Solve for x: Now, we have 8x = -10. To isolate x, we need to divide both sides of the equation by 8. This will give us the value of x that satisfies the equation: 8x / 8 = -10 / 8. This simplifies to x = -10/8.

  4. Simplify the fraction: We're not quite done yet! We can simplify the fraction -10/8 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us: x = -5/4.

  5. Convert to decimal form (optional): Sometimes, it's helpful to express the root as a decimal. To do this, simply divide -5 by 4: x = -1.25.

And there you have it! We've successfully found the root of the function h(x) = 8x + 10. The value of x that makes h(x) equal to zero is -1.25.

Why is -1.25 the Correct Answer?

Now, let's take a moment to understand why -1.25 is indeed the correct answer. Remember, the root of a function is the x-value that makes the function equal to zero. So, to verify our answer, we can plug x = -1.25 back into the original function and see if we get zero:

h(x) = 8x + 10

h(-1.25) = 8(-1.25) + 10

h(-1.25) = -10 + 10

h(-1.25) = 0

As you can see, when we substitute x = -1.25 into the function, we get an output of zero. This confirms that -1.25 is indeed the root of the function. We did it!

Another way to visualize this is by looking at the graph of the function h(x) = 8x + 10. The graph is a straight line, and the root (-1.25) is the point where the line crosses the x-axis. If you were to plot the graph, you'd see that the line intersects the x-axis at x = -1.25. This provides a visual confirmation of our solution.

Common Mistakes to Avoid

Finding the roots of a function is a pretty straightforward process, but there are a few common mistakes that students sometimes make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer.

  • Incorrectly isolating x: This is probably the most common mistake. Remember, the goal is to get x by itself on one side of the equation. This means performing the correct operations (addition, subtraction, multiplication, division) in the correct order. For example, if you forget to subtract 10 from both sides of the equation in our example, you'll end up with the wrong answer. Always double-check your steps!
  • Sign errors: Sign errors are another frequent culprit. When moving terms from one side of the equation to the other, remember to change their signs. For instance, when we subtracted 10 from both sides of 8x + 10 = 0, the +10 became -10 on the right side. Pay close attention to positive and negative signs!
  • Not simplifying the fraction: While x = -10/8 is technically correct, it's best to simplify it to x = -5/4 or x = -1.25. Simplifying fractions makes the answer cleaner and easier to work with. Always simplify your answers as much as possible!
  • Forgetting to check the answer: As we demonstrated earlier, plugging your solution back into the original equation is a great way to verify that you've done everything correctly. This is a crucial step that can save you from making mistakes. Always check your answer!

By being mindful of these common mistakes, you can significantly improve your accuracy when finding the roots of functions.

Practice Makes Perfect: More Examples

The best way to master any mathematical concept is through practice. So, let's work through a couple more examples to solidify your understanding of finding roots. We'll use the same step-by-step process we used for h(x) = 8x + 10, so you can see how it applies to different functions.

Example 1: Find the roots of f(x) = 2x - 6

  1. Set the function equal to zero: 2x - 6 = 0
  2. Isolate the term with x: Add 6 to both sides: 2x = 6
  3. Solve for x: Divide both sides by 2: x = 3
  4. Check the answer: f(3) = 2(3) - 6 = 6 - 6 = 0. So, the root is x = 3.

Example 2: Find the roots of g(x) = -3x + 9

  1. Set the function equal to zero: -3x + 9 = 0
  2. Isolate the term with x: Subtract 9 from both sides: -3x = -9
  3. Solve for x: Divide both sides by -3: x = 3
  4. Check the answer: g(3) = -3(3) + 9 = -9 + 9 = 0. So, the root is x = 3.

As you can see, the process is the same regardless of the specific function. Just remember to follow the steps carefully and double-check your work. The more you practice, the more confident you'll become in your ability to find roots!

Conclusion: You've Got This!

Finding the roots of a function might have seemed daunting at first, but hopefully, you now have a clear understanding of the process. We've covered the definition of roots, the step-by-step method for finding them, common mistakes to avoid, and even worked through some examples. Remember, the key is to practice and apply what you've learned.

So, the next time you encounter a function and need to find its roots, don't be intimidated! Just follow the steps, stay organized, and you'll be able to solve it with confidence. Keep practicing, keep learning, and you'll be a root-finding pro in no time! You guys got this!