Quadratic Equation: Finding The Coefficient A

Hey guys! Let's dive into a fun math problem today. We're going to break down a quadratic equation and figure out a specific coefficient. It might sound intimidating, but trust me, we'll take it step by step and make it super clear. So, let's jump right into it!

Understanding the Problem

So, our friend Isoke is tackling the quadratic equation $10x^2 + 40x - 13 = 0$ by using a method called completing the square. This is a classic technique for solving quadratic equations, and it involves manipulating the equation to get it into a form where we can easily isolate x. Isoke has already started the process and has reached a particular step: $A(x^2 + 4x) = 13$. Our mission, should we choose to accept it (and we do!), is to find the value of A. This value, A, is a coefficient, which is just a fancy term for the number that's multiplying the expression inside the parentheses, which is $(x^2 + 4x)$ in this case. To really nail this, we need to understand how Isoke got to this step, and that involves retracing her steps, which will naturally lead us to discovering the value of A. Remember, guys, math is like a puzzle, and each step is a clue!

Breaking Down the Initial Steps

Let's rewind a bit and look at the original equation: $10x^2 + 40x - 13 = 0$. The first thing Isoke likely did was to isolate the terms containing x on one side of the equation. She did this by adding 13 to both sides. This gives us: $10x^2 + 40x = 13$. This move is crucial because it sets the stage for the next step: factoring out a common factor. This common factor is precisely what we're looking for, the value of A. Factoring out a common factor simplifies the equation and helps us get closer to completing the square. Now, let's think about what number we can pull out from both $10x^2$ and $40x$. What's the greatest common factor, guys? If you guessed 10, you're on the right track! This is a key step in identifying A and understanding the transformation of the equation.

Identifying the Coefficient A

Now comes the moment of truth! We've got the equation $10x^2 + 40x = 13$. Isoke then factored out the common factor from the left side. As we discussed, the greatest common factor of $10x^2$ and $40x$ is 10. So, she factored out 10, which gives us: $10(x^2 + 4x) = 13$. Boom! Look familiar? This exactly matches the form $A(x^2 + 4x) = 13$. So, by comparing the two equations, it's crystal clear that the value of A is 10. See? We solved it! This factoring step is super important in completing the square. It allows us to manipulate the equation into a form that's easier to work with. Plus, it directly reveals the value of A, which is what we were after all along. Give yourselves a pat on the back, guys, we're making progress!

Completing the Square: A Quick Overview

Now that we've found A, let's take a quick detour and zoom out to understand the bigger picture: completing the square. This technique is a powerful way to solve quadratic equations, and it's worth understanding the general idea. The goal of completing the square is to rewrite the quadratic equation in the form $(x + p)^2 = q$, where p and q are constants. This form is incredibly useful because we can then easily solve for x by taking the square root of both sides. It's like magic, but it's actually just clever algebra! To get to this form, we need to add a specific constant to both sides of the equation. This constant is carefully chosen so that the expression on the left side becomes a perfect square trinomial. This is where the "completing the square" part comes in. We're essentially adding the missing piece to make a perfect square. It might sound complicated, but once you've done it a few times, it becomes second nature. Think of it as a recipe, guys, follow the steps, and you'll bake a perfect quadratic solution!

Continuing the Process with A = 10

Okay, let's bring it back to our specific problem. We know A is 10, and we have the equation $10(x^2 + 4x) = 13$. To complete the square, we focus on the expression inside the parentheses: $x^2 + 4x$. We need to figure out what constant to add to this expression to make it a perfect square trinomial. Remember the perfect square trinomial pattern: $(x + b)^2 = x^2 + 2bx + b^2$. Comparing this pattern to our expression, we see that $2b = 4$, so $b = 2$. Therefore, the constant we need to add is $b^2 = 2^2 = 4$. However, here's a crucial detail: we're not just adding 4 to the left side of the equation. We're adding 4 inside the parentheses, which are being multiplied by 10. So, we're actually adding $10 * 4 = 40$ to the left side. This means we also need to add 40 to the right side to keep the equation balanced. This is a common pitfall, guys, so always remember to account for any coefficients outside the parentheses! Keeping the equation balanced is like maintaining equilibrium on a seesaw – if you add weight on one side, you need to add the same weight to the other side.

The Next Steps in Solving for x

Adding 40 to both sides of our equation, $10(x^2 + 4x) = 13$, gives us: $10(x^2 + 4x + 4) = 13 + 40$, which simplifies to $10(x^2 + 4x + 4) = 53$. Now, the expression inside the parentheses is a perfect square trinomial! We can rewrite it as $(x + 2)^2$. So, our equation becomes: $10(x + 2)^2 = 53$. See how we're getting closer to isolating x? We're on the home stretch! The next step is to divide both sides by 10, which gives us: $(x + 2)^2 = 5.3$. Now, we can take the square root of both sides, being sure to consider both the positive and negative roots. This will give us two possible values for $x + 2$. Finally, we can subtract 2 from both sides to solve for x. And there you have it, guys, we've walked through the process of completing the square and found the solutions for x! It's like climbing a mountain – each step brings you closer to the summit, and the view from the top is totally worth it!

Why Completing the Square Matters

You might be wondering, why bother with completing the square when we have other methods for solving quadratic equations, like the quadratic formula? That's a valid question, guys! While the quadratic formula is a trusty tool, completing the square provides a deeper understanding of the structure of quadratic equations. It reveals how the equation can be transformed and manipulated, which can be incredibly useful in various mathematical contexts. Plus, completing the square is the foundation for deriving the quadratic formula itself! So, it's like understanding the ingredients and recipe before you bake the cake. Knowing the process helps you appreciate the result even more. Moreover, completing the square has applications beyond just solving equations. It's used in calculus, conic sections, and other areas of mathematics. So, mastering this technique expands your mathematical toolkit and makes you a more versatile problem-solver. Think of it as learning a new language – the more languages you speak, the more conversations you can have!

Conclusion

So, there you have it! We successfully found the value of A in Isoke's quadratic equation problem. By understanding the process of completing the square and retracing the initial steps, we were able to identify that A equals 10. We also took a broader look at completing the square and why it's such a valuable technique. Remember, guys, math is a journey of discovery, and each problem we solve adds to our understanding and skills. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!