Finding The Equivalent Equation For The Graph Of Y=x^2+11x+24

by ADMIN 62 views

Hey everyone! Today, we're diving deep into the fascinating world of quadratic equations and their graphical representations. Specifically, we're going to explore the equation y = x² + 11x + 24 and figure out which of the given options represents the same graph. It's like a puzzle, and we're the detectives! So, grab your thinking caps, and let's get started on this mathematical adventure.

Understanding the Quadratic Equation

Before we jump into the specifics, let's make sure we're all on the same page about quadratic equations. A quadratic equation is essentially a polynomial equation of the second degree. It generally takes the form of y = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic equation is a parabola, a U-shaped curve that opens either upwards or downwards depending on the sign of the coefficient a. In our case, the equation y = x² + 11x + 24 fits this form perfectly, with a being 1, b being 11, and c being 24. The solutions to the quadratic equation, also known as the roots or zeros, are the points where the parabola intersects the x-axis (where y equals 0). These roots play a crucial role in determining the equivalent factored form of the equation, which is what we're after.

Now, the beauty of quadratic equations lies in their flexibility. They can be expressed in different forms, each offering a unique perspective. The standard form, which we have (y = x² + 11x + 24), is great for identifying the coefficients a, b, and c. However, to find the equivalent equation that represents the same graph, we need to delve into the factored form. The factored form of a quadratic equation looks like this: y = (x + p)(x + q), where p and q are the roots of the equation. When we expand this factored form, we get y = x² + (p + q)x + pq. Notice how the coefficient of the x term is the sum of the roots (p + q), and the constant term is the product of the roots (pq). This relationship is the key to unlocking our puzzle. It allows us to work backward from the standard form to the factored form, and vice versa.

Think of it like this: the standard form presents the equation in a ready-to-graph format, while the factored form reveals the equation's fundamental building blocks – its roots. By understanding the connection between these forms, we can effortlessly transform one into the other, and ultimately, identify which equation is equivalent to y = x² + 11x + 24. So, with this knowledge in our arsenal, let's move on to the exciting part – factoring our equation and finding the answer!

Factoring the Quadratic Equation

Okay, guys, let's roll up our sleeves and get our hands dirty with some factoring! This is where the real magic happens. Factoring a quadratic equation means breaking it down into its constituent parts – the factors that, when multiplied together, give us the original equation. In our case, we want to express y = x² + 11x + 24 in the factored form (x + p)(x + q). As we discussed earlier, this means finding two numbers, p and q, that satisfy two conditions: their sum must equal the coefficient of the x term (which is 11), and their product must equal the constant term (which is 24). Finding these two numbers is like solving a mini-puzzle within the larger puzzle, but don't worry, we've got this!

Let's start by thinking about the factors of 24. We need pairs of numbers that multiply to 24. These pairs are: 1 and 24, 2 and 12, 3 and 8, and 4 and 6. Now, out of these pairs, which one adds up to 11? Bingo! It's 3 and 8. So, p and q are 3 and 8 (or vice versa – the order doesn't matter in multiplication). This means we can rewrite our equation y = x² + 11x + 24 in the factored form as y = (x + 3)(x + 8). See? It's not as daunting as it might seem at first. By systematically considering the factors of the constant term and checking their sum, we've successfully unlocked the factored form of our equation.

Now, you might be wondering, why is this factored form so important? Well, it's the key to identifying the equivalent equation from the given options. The factored form directly reveals the roots of the equation, which in turn define the parabola's shape and position on the graph. Equations with the same roots will have the same graph, and that's precisely what we're looking for. So, with our factored form y = (x + 3)(x + 8) in hand, we're ready to compare it with the given options and pinpoint the one that matches. This is like the final piece of the puzzle snapping into place, and it's incredibly satisfying when it happens. Let's move on to the next step and see which option is our winner!

Comparing with the Options

Alright, the moment of truth has arrived! We've successfully factored our equation into y = (x + 3)(x + 8). Now, let's put on our detective hats again and carefully compare this factored form with the options provided. Remember, we're looking for the equation that is equivalent to our original equation, meaning it represents the same parabola on the graph. This means it must have the same roots, and therefore, the same factored form.

The options we have are:

  • y = (x + 8)(x + 3)
  • y = (x + 4)(x + 6)
  • y = (x + 9)(x + 2)
  • y = (x + 7)(x + 4)

Let's go through them one by one and see which one matches our factored form, y = (x + 3)(x + 8). The first option, y = (x + 8)(x + 3), looks awfully familiar, doesn't it? In fact, it's exactly the same as our factored form! Remember, the order in which we multiply factors doesn't matter (this is the commutative property of multiplication). So, (x + 3)(x + 8) is the same as (x + 8)(x + 3). We have a winner! Option 1 is the equivalent equation we've been searching for.

But just for thoroughness, let's quickly examine the other options to make sure they don't match. Option 2, y = (x + 4)(x + 6), has roots of -4 and -6, which are different from our roots of -3 and -8. Similarly, Option 3, y = (x + 9)(x + 2), has roots of -9 and -2, and Option 4, y = (x + 7)(x + 4), has roots of -7 and -4. None of these match our roots, so they cannot be equivalent to our original equation. This confirms that our initial finding was correct: Option 1 is the only equation that represents the same graph as y = x² + 11x + 24.

So, there you have it! We've successfully navigated the world of quadratic equations, factored a tricky expression, and pinpointed the equivalent equation from a set of options. It's like solving a complex riddle, and the feeling of accomplishment is fantastic. But the journey doesn't end here. Understanding these concepts opens the door to even more exciting mathematical explorations. Keep practicing, keep exploring, and you'll be amazed at what you can achieve!

Conclusion

Awesome job, guys! We've reached the end of our quest to find the equivalent equation for the graph of y = x² + 11x + 24. Through the power of factoring and careful comparison, we've confidently identified that y = (x + 8)(x + 3) is the correct answer. This journey has reinforced our understanding of quadratic equations, factored forms, and the relationship between roots and graphs. Remember, the key to mastering these concepts is practice and exploration. So, keep tackling those mathematical challenges, and you'll become a true equation-solving pro! And most importantly, have fun along the way!