Solving Quadratic Equations: Find A And B

Hey math enthusiasts! Today, we're diving into a cool problem that combines algebra and a bit of puzzle-solving. We're going to explore the equation $x^2 - x + 5 = (x - a)^2 + b$. Our mission, should we choose to accept it, is to figure out the values of a and b. These values are super important because they let us rewrite the quadratic equation in a more useful form. This technique is often used in completing the square, a powerful tool for solving quadratic equations, understanding their graphs, and tackling more advanced math concepts. Ready to jump in? Let's go!

Understanding the Core Problem

So, what's this equation all about? We've got a quadratic expression on the left side: $x^2 - x + 5$. This is a standard quadratic, meaning it has an $x^2$ term, an $x$ term, and a constant. On the right side, we have $(x - a)^2 + b$. This is where things get interesting. Notice how $(x - a)^2$ looks? That's a perfect square. When we expand it, we'll get another quadratic, but this one is in a special form, often called the vertex form of a quadratic equation. This form is incredibly useful because it directly reveals the vertex of the parabola (the U-shaped graph) that the equation represents. The vertex is like the turning point of the parabola, and knowing its coordinates gives us valuable insights into the function's behavior. The constant b simply shifts the parabola up or down along the y-axis.

Our task is to manipulate the left side of the equation, $x^2 - x + 5$, to match the form of the right side, $(x - a)^2 + b$. We'll do this using a technique called completing the square. By rewriting the equation in this vertex form, we can easily identify the values of a and b and understand the transformation that's been applied to the basic $x^2$ function. This is more than just finding numbers; it's about understanding how quadratics work and how we can use different forms of the same equation to solve problems and visualize the function graphically. This understanding is foundational for more advanced topics in algebra and calculus.

Now, let's break down the process step by step, making sure everything is clear and easy to follow. We're going to work through the algebra, showing all the important steps, so that you can follow along easily. This isn't just about getting an answer; it's about understanding why each step works. This will help you to solve similar problems on your own. It's like learning a recipe – once you understand the ingredients and how they interact, you can start making your own variations.

Unveiling the Values: Finding 'a'

Alright, let's get our hands dirty and figure out those elusive values of a and b. The first step is to focus on the term $(x - a)^2$. We need to figure out what value of a makes the left side of our equation, $x^2 - x + 5$, look like a perfect square plus a constant. Remember, a perfect square is an expression that can be written as $(x - something)^2$. To do this, we'll use the technique of completing the square. It's like a magic trick, but instead of rabbits, we pull out simplified equations! The trick is to take the coefficient of the x term (which is -1 in our case), divide it by 2 (giving us -1/2), and then square the result ((-1/2)^2 = 1/4).

Let's apply this process. We start with our original equation and manipulate it: $x^2 - x + 5 = (x - a)^2 + b$. We look at the $x^2 - x$ part and try to form a perfect square. To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of the x term is -1. Half of -1 is -1/2. Squaring -1/2 gives us 1/4. Now we rewrite the equation like so: $x^2 - x + 1/4 - 1/4 + 5 = (x - a)^2 + b$. Now we group the first three terms as a perfect square: $(x - 1/2)^2 - 1/4 + 5 = (x - a)^2 + b$. See that? $(x - 1/2)^2$ is our perfect square. This is the beauty of completing the square! Now we can easily see that $a = 1/2$. So, the first part of our mystery is solved! We have identified the value of a by matching the coefficients in the expanded form of $(x - a)^2$ and the original quadratic equation. Remember, matching the coefficients is the key! This understanding not only helps solve the problem at hand but also builds a strong foundation for tackling more complex algebraic challenges.

By following these steps, you've not only solved for a, but also gained valuable experience with completing the square, a key skill in algebra. The most important thing is to understand the logic behind the method. This technique will be extremely useful in future studies, so take your time to digest the steps and the math behind it. This is like understanding the foundation of a house. The better the foundation, the stronger the house will be. Understanding the foundation of math will make everything else much easier.

Pinpointing 'b': The Final Piece

Now that we've found a, let's nail down b. We have our equation in this form: $(x - 1/2)^2 - 1/4 + 5 = (x - a)^2 + b$. Since we've already determined that $a = 1/2$, we can rewrite the equation as: $(x - 1/2)^2 - 1/4 + 5 = (x - 1/2)^2 + b$. Our goal is to isolate b and find its numerical value. Notice that the $(x - 1/2)^2$ terms appear on both sides of the equation. This is a good sign that things are falling into place!

To find b, we need to simplify the constant terms on the left side of the equation. We have $-1/4 + 5$. To add these, we can express 5 as a fraction with a denominator of 4. So, 5 becomes 20/4. Now we can add the fractions: $-1/4 + 20/4 = 19/4$. Thus, the left side simplifies to $(x - 1/2)^2 + 19/4$. Now we have: $(x - 1/2)^2 + 19/4 = (x - 1/2)^2 + b$. By comparing the two sides, we can clearly see that $b = 19/4$.

And there you have it! We've found both a and b: $a = 1/2$ and $b = 19/4$. This means we have successfully rewritten the original equation $x^2 - x + 5 = (x - a)^2 + b$ in the vertex form: $x^2 - x + 5 = (x - 1/2)^2 + 19/4$. Congratulations! You have learned to find the values of a and b by completing the square and understanding the vertex form of a quadratic equation. This skill is extremely valuable for understanding quadratics and is a fundamental concept in algebra. This isn't just about solving a single equation. You've now gained a deeper insight into quadratic equations and the power of algebraic manipulation. These are skills that will serve you well in future math courses and in solving real-world problems. Keep practicing, and these concepts will become second nature to you. It's like any skill - the more you practice, the better you become!

Expanding the Knowledge: The Vertex Form

Let's take a moment to really appreciate the significance of what we've accomplished. By finding a and b and rewriting the equation in the form $(x - a)^2 + b$, we've effectively transformed our quadratic into its vertex form. The vertex form of a quadratic equation is a powerful tool because it directly reveals the vertex of the parabola. The general vertex form is $y = k(x - h)^2 + k$, where (h, k) is the vertex of the parabola. In our case, the vertex is at the point $(1/2, 19/4)$. Understanding the vertex is crucial because it provides key information about the graph of the equation. The vertex represents either the minimum or maximum point of the parabola, depending on whether the coefficient of the $x^2$ term is positive or negative.

Since the coefficient of the $x^2$ term in our original equation, $x^2 - x + 5$, is positive, the parabola opens upwards, and the vertex is its minimum point. This means that the lowest point on the graph of this equation is at the coordinates $(1/2, 19/4)$. The value of a tells us how far the parabola is shifted horizontally, and the value of b tells us how far it's shifted vertically. Thus, we have a complete picture of the parabola's position in the coordinate plane. Think of it like this: the vertex form is like having a map that tells you exactly where the parabola is located. You can determine not only the location of the minimum or maximum point, but also how the parabola is transformed from the basic parabola $y = x^2$. This is a powerful analytical tool, and a crucial skill in the study of quadratics. This understanding also serves as a gateway to more advanced mathematical concepts, and is the foundation for topics like calculus.

So, as you can see, the ability to rewrite a quadratic equation into its vertex form provides a wealth of information about its graph and its behavior. Now, you can quickly sketch the graph of this equation. You know it's a parabola that opens upwards, with its lowest point at $(1/2, 19/4)$. You have all the information you need to analyze the graph! You have successfully completed the problem and also expanded your knowledge of quadratic equations and their graphs. Keep up the good work and continue to explore the fascinating world of mathematics!

Practice Makes Perfect: Additional Examples

Now that you've grasped the core concepts, let's try some practice problems to solidify your understanding. The best way to master any mathematical concept is through practice. The more you work through different examples, the more comfortable you'll become with the process and the more adept you'll be at recognizing patterns and applying the techniques. Remember, the key is to practice consistently and to focus on understanding the underlying logic of each step.

Here's an example: Find the values of a and b for the equation $x^2 + 4x + 7 = (x - a)^2 + b$. Follow these steps to complete the square: Take the coefficient of the x term (which is 4), divide it by 2 (resulting in 2), and square the result (2^2 = 4). Add and subtract this value to the left side: $x^2 + 4x + 4 - 4 + 7$. Now, group the perfect square: $(x + 2)^2 - 4 + 7$. Simplify the constant terms: $(x + 2)^2 + 3$. From this, we can see that the equation $(x^2 + 4x + 7)$ in the form $(x - a)^2 + b$ is $(x + 2)^2 + 3$. Thus, a = -2 and b = 3. Notice the sign change when comparing $(x + 2)^2$ to $(x - a)^2$. It’s always important to pay attention to these details.

Now, here is another example: Determine a and b for the equation $x^2 - 6x + 2 = (x - a)^2 + b$. Applying our procedure, divide the coefficient of the x term (-6) by 2, which gives us -3. Square -3 to get 9. Rewrite the equation: $x^2 - 6x + 9 - 9 + 2$. Form the perfect square: $(x - 3)^2 - 9 + 2$. Simplify: $(x - 3)^2 - 7$. Comparing this to the form $(x - a)^2 + b$, we find that a = 3 and b = -7. This further reinforces the importance of consistent practice and attention to detail. This method helps you gain confidence and provides you with the skills to address a broad range of related problems. By practicing these types of problems, you’ll be ready for more challenging questions.

Keep practicing, and don't be afraid to ask for help or review the steps when needed. Solving these types of problems will become increasingly easy as you become more familiar with the process. Like any skill, practice makes perfect. So, keep practicing, and you will become skilled at finding the values of a and b in no time! Remember, consistency and focused practice will help you achieve mastery.