Expanding And Simplifying: 40(y+1)^2 - 9(x-3)^2 = 360

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Expanding and Simplifying: 40(y+1)^2 - 9(x-3)^2 = 360

Hey guys! Today, we're diving into a fun math problem where we'll expand and simplify the equation 40(y+1)^2 - 9(x-3)^2 = 360. This might look a bit intimidating at first, but trust me, we'll break it down step by step. We'll be using some basic algebraic principles and a bit of patience. So, grab your pencils, and let's get started!

Understanding the Equation

Before we jump into expanding and simplifying, let's take a good look at the equation: 40(y+1)^2 - 9(x-3)^2 = 360. Notice that we have two squared terms, (y+1)^2 and (x-3)^2. These suggest that we might be dealing with a conic section, specifically a hyperbola. However, our primary goal here is to expand and simplify the equation, so we'll focus on that for now. Recognizing the potential form helps in understanding the final result and how it fits into broader mathematical concepts, but it's not essential for the simplification process itself. The key is to methodically apply algebraic rules to transform the equation into a more manageable and understandable form. Remember, simplification often reveals hidden structures and relationships within the equation, which can be crucial for further analysis or problem-solving.

Breaking Down the Components

To expand this equation effectively, we need to tackle each part separately. The equation has two main components: 40(y+1)^2 and -9(x-3)^2. We'll start by expanding each of these individually before combining them and simplifying the entire equation. This approach helps in managing the complexity and reduces the chances of making errors. Think of it like dismantling a machine into smaller parts to understand how each part works before reassembling it. Each term requires careful application of the distributive property and the rules of exponents. The constants (40 and -9) will be multiplied after the squared terms are expanded. By addressing each component methodically, we can ensure accuracy and clarity in our simplification process.

Step-by-Step Expansion

Let's begin by expanding the first term, 40(y+1)^2. Remember, (y+1)^2 means (y+1) multiplied by itself. So, the first step is to expand (y+1)(y+1). To do this, we'll use the FOIL method (First, Outer, Inner, Last). First, we multiply the first terms (y * y = y^2). Outer terms (y * 1 = y). Inner terms (1 * y = y). And Last terms (1 * 1 = 1). Combining these gives us y^2 + y + y + 1, which simplifies to y^2 + 2y + 1. Now, we need to multiply this entire expression by 40. So, 40 * (y^2 + 2y + 1) becomes 40y^2 + 80y + 40. This completes the expansion of the first term.

Expanding the First Term: 40(y+1)^2

The detailed breakdown of expanding 40(y+1)^2 is as follows:

  1. Expand (y+1)^2: This means (y+1) * (y+1). Using the FOIL method, we get:

    • First: y * y = y^2
    • Outer: y * 1 = y
    • Inner: 1 * y = y
    • Last: 1 * 1 = 1

    Combining these terms, we have y^2 + y + y + 1, which simplifies to y^2 + 2y + 1.

  2. Multiply by 40: Now, we multiply the entire expression by 40:40 * (y^2 + 2y + 1) = 40y^2 + 80y + 40.

This completes the expansion of the first term. Each step is crucial to ensure accuracy, and the FOIL method helps us systematically expand the binomial product. The resulting quadratic expression is a key component in the overall equation.

Expanding the Second Term: -9(x-3)^2

Now, let's move on to the second term, -9(x-3)^2. This is similar to the first term, but we need to pay close attention to the negative sign and the subtraction inside the parenthesis. First, we expand (x-3)^2, which means (x-3) * (x-3). Again, using the FOIL method: First (x * x = x^2), Outer (x * -3 = -3x), Inner (-3 * x = -3x), Last (-3 * -3 = 9). Combining these gives us x^2 - 3x - 3x + 9, which simplifies to x^2 - 6x + 9. Now, we multiply this entire expression by -9. So, -9 * (x^2 - 6x + 9) becomes -9x^2 + 54x - 81. Remember that multiplying by a negative number changes the sign of each term inside the parenthesis. This completes the expansion of the second term.

The detailed breakdown of expanding -9(x-3)^2 is as follows:

  1. Expand (x-3)^2: This means (x-3) * (x-3). Using the FOIL method, we get:

    • First: x * x = x^2
    • Outer: x * -3 = -3x
    • Inner: -3 * x = -3x
    • Last: -3 * -3 = 9

    Combining these terms, we have x^2 - 3x - 3x + 9, which simplifies to x^2 - 6x + 9.

  2. Multiply by -9: Now, we multiply the entire expression by -9: -9 * (x^2 - 6x + 9) = -9x^2 + 54x - 81.

This completes the expansion of the second term. Note the importance of distributing the negative sign correctly to ensure the signs of the resulting terms are accurate.

Combining and Simplifying

Now that we've expanded both terms, let's combine them. We have 40y^2 + 80y + 40 from the first term and -9x^2 + 54x - 81 from the second term. Our equation now looks like this: 40y^2 + 80y + 40 - 9x^2 + 54x - 81 = 360. The next step is to combine like terms. We have the constant terms 40 and -81, which combine to -41. So, the equation becomes 40y^2 + 80y - 9x^2 + 54x - 41 = 360. To further simplify, we'll move the 360 from the right side to the left side by subtracting 360 from both sides. This gives us 40y^2 + 80y - 9x^2 + 54x - 41 - 360 = 0, which simplifies to 40y^2 + 80y - 9x^2 + 54x - 401 = 0. This is the simplified form of the equation.

Putting It All Together

To recap, we've expanded and simplified the equation 40(y+1)^2 - 9(x-3)^2 = 360 through these steps:

  1. Expanded 40(y+1)^2 to get 40y^2 + 80y + 40.
  2. Expanded -9(x-3)^2 to get -9x^2 + 54x - 81.
  3. Combined the expanded terms: 40y^2 + 80y + 40 - 9x^2 + 54x - 81 = 360.
  4. Simplified by combining like terms: 40y^2 + 80y - 9x^2 + 54x - 41 = 360.
  5. Moved the constant to the left side: 40y^2 + 80y - 9x^2 + 54x - 41 - 360 = 0.
  6. Final Simplified Equation: 40y^2 + 80y - 9x^2 + 54x - 401 = 0.

Each step involved careful application of algebraic principles, such as the FOIL method and combining like terms. The final simplified equation represents the same relationship as the original equation but in a more manageable form.

Final Simplified Equation

The final simplified form of the equation 40(y+1)^2 - 9(x-3)^2 = 360 is 40y^2 + 80y - 9x^2 + 54x - 401 = 0. This equation is now in a general form, which can be further analyzed to identify the conic section it represents (in this case, a hyperbola) and to extract other important information, such as the center, vertices, and asymptotes. Simplifying equations like this is a fundamental skill in algebra and is crucial for solving more complex problems in mathematics and other fields. The process of expanding and simplifying allows us to transform an equation into a form that is easier to work with and understand. Keep practicing, and you'll become a pro at this in no time!

Understanding the Result

So, what does this final equation, 40y^2 + 80y - 9x^2 + 54x - 401 = 0, tell us? Well, it represents a hyperbola. The presence of both y^2 and x^2 terms with opposite signs (one positive and one negative) is a key indicator of a hyperbola. This form is the general form of a conic section, and from here, we could potentially complete the square for both the x and y terms to get the equation into standard form. The standard form would give us more immediate insights into the hyperbola's properties, such as its center, axes, and asymptotes. While we've achieved our primary goal of expanding and simplifying, recognizing the type of conic section provides a deeper understanding of the equation's graphical representation and behavior. This connection between algebraic form and geometric shape is a fundamental concept in analytic geometry.

Conclusion

Alright guys, we've successfully expanded and simplified the equation 40(y+1)^2 - 9(x-3)^2 = 360! We started with a complex-looking equation and, by systematically applying algebraic principles, transformed it into a more manageable form: 40y^2 + 80y - 9x^2 + 54x - 401 = 0. Remember, the key to solving these problems is to break them down into smaller, more manageable steps. Expand each term individually, combine like terms, and don't forget to double-check your work along the way. This process not only simplifies the equation but also gives us a deeper understanding of its structure and the relationships between its components. Keep practicing, and you'll be able to tackle even the most challenging equations with confidence. Remember, math is like a puzzle, and each step we take brings us closer to the solution. You've got this!