Positive Values Of The Expression: Unveiling X & Y

by SLV Team 51 views
Positive Values of the Expression: Unveiling x & y

Hey math enthusiasts! Today, we're diving into an interesting problem involving algebraic expressions. We want to find out for which values of x and y the expression ((x^2+y^2)/xy) / ((x^3+y^3)/xy) results in positive values. Sounds fun, right? Let's break it down and see how we can solve this together. This is a classic example of how understanding algebraic manipulations and the properties of numbers can help us solve seemingly complex problems. We'll need to use some basic algebraic principles, paying close attention to the conditions that make the expression positive. This involves understanding fractions, factoring, and the signs of numbers. I'm excited to explore this with you guys, so let's get started!

This exploration will require us to carefully consider the conditions under which a fraction is positive. Remember, a fraction is positive when both the numerator and the denominator are positive, or when both are negative. Also, we have to consider the restrictions on x and y due to the presence of xy and (x^3 + y^3) in the denominator. These are crucial elements to take into account. We'll start by simplifying the given expression to make it easier to work with. Then, we will consider the signs of the numerator and the denominator to determine the values of x and y that satisfy the required conditions. It's all about methodically working through the steps, so we can arrive at the right answer. We will carefully analyze this. Let’s get into the details, shall we?

First things first, let's simplify the original expression. We have ((x^2+y^2)/xy) / ((x^3+y^3)/xy). When dividing fractions, we can multiply by the reciprocal of the second fraction. This simplifies our expression to ((x^2 + y^2) / xy) * (xy / (x^3 + y^3)). Notice that the xy terms cancel out, leaving us with (x^2 + y^2) / (x^3 + y^3). Much simpler, right? Now, our focus shifts to determining the conditions where (x^2 + y^2) / (x^3 + y^3) is positive. Let's not forget the initial condition: x and y cannot be zero, as they are in the denominators of the original expression. We'll have to keep this in mind as we proceed. We are now closer to solving the problem, and we've already done the first step which is simplifying the expression. Let's make sure we clearly understand the next steps as we will now analyze the simplified expression.

Now that we've simplified the expression, let's dig deeper into when (x^2 + y^2) / (x^3 + y^3) is positive. For a fraction to be positive, either both the numerator and denominator are positive, or both are negative. Let's consider each case separately. First, consider the case where both the numerator and denominator are positive. The numerator is x^2 + y^2. Since squares of real numbers are always non-negative, and the sum of two non-negative numbers is non-negative, x^2 + y^2 is always greater than or equal to zero. But, since x and y can't both be zero (because of the original denominator), x^2 + y^2 is strictly positive unless x and y are both zero. So, if x^2 + y^2 is positive, we also need x^3 + y^3 to be positive for the entire fraction to be positive. Next, let’s consider the case when both numerator and denominator are negative. But the numerator x^2 + y^2 can never be negative. So we only need to focus on when both are positive.

Analyzing Positive Values: Delving into Numerator and Denominator

Let’s zoom in on the case where both the numerator and the denominator are positive. We already know that x^2 + y^2 is positive as long as x and y are not both zero. The next important part is to focus on the denominator, which is x^3 + y^3. So, we need to find the conditions where x^3 + y^3 is positive. Now, we can factor x^3 + y^3 using the sum of cubes formula: x^3 + y^3 = (x + y)(x^2 - xy + y^2). For x^3 + y^3 to be positive, we need (x + y)(x^2 - xy + y^2) to be positive. This can happen in two scenarios: either both factors are positive, or both factors are negative. This is where things get interesting. Notice that the factor x^2 - xy + y^2 can be rewritten as (x - y/2)^2 + (3/4)y^2. This is always greater than or equal to zero. It's important because it tells us that x^2 - xy + y^2 is always non-negative. This is super useful to know. If x^2 - xy + y^2 is zero, it means that x = y = 0, which we already know is impossible.

So, for (x + y)(x^2 - xy + y^2) to be positive, and given that x^2 - xy + y^2 is always non-negative, then it means that x + y must be positive. This implies that x + y > 0. Also, since the original expression contained xy in the denominator, x and y cannot be zero. These conditions together give us the solution. Essentially, the condition that makes the original expression positive is that x + y > 0, and neither x nor y can be zero. We're getting closer to our final answer. Understanding how to factor and analyze these algebraic expressions is key. Now that we've simplified everything, let's explore this more. We are now able to determine when the initial expression has positive values.

Putting It All Together: Finding the Solution

Alright, guys! Let's summarize and put everything together. We've simplified the expression to (x^2 + y^2) / (x^3 + y^3). We know that x^2 + y^2 is always positive (unless x and y are both zero, which is not allowed). To ensure the overall expression is positive, the denominator x^3 + y^3 must also be positive. We factored x^3 + y^3 into (x + y)(x^2 - xy + y^2). The term x^2 - xy + y^2 is always non-negative. Therefore, for (x + y)(x^2 - xy + y^2) to be positive, the term x + y must be positive as well. So, the key condition for the original expression to be positive is x + y > 0, and also, neither x nor y can be zero. This means that x and y can be any real numbers that satisfy these conditions. We've managed to unravel the problem, step by step! This demonstrates the beauty of algebra; it allows us to break down complicated-looking expressions into manageable parts, and solve the problem systematically. The importance of the restrictions on x and y cannot be emphasized enough. We have now arrived at the final answer. Let's make sure we clearly understand the final answer before moving on.

So, the final answer to the question: "For what values of x and y does the expression ((x^2+y^2)/xy) / ((x^3+y^3)/xy) yield positive function values?" is:

  • Condition: x + y > 0
  • Restriction: x ≠ 0 and y ≠ 0

That's it, we're done! We've successfully navigated through the expression, simplified it, factored it, and identified the exact conditions that make the expression positive. I hope you guys enjoyed this exploration. Keep practicing, and you'll become pros at these kinds of problems in no time. If you have any questions or want to explore similar problems, let me know. Happy math-ing!