Find The Missing Term In The Perfect Square Trinomial 1/9 X^(2) - 10/3 X^(2 )y

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Hey guys! Ever stumbled upon a math problem that looks like a puzzle? Well, today we're diving into one of those – a perfect square trinomial with a missing piece. Specifically, we're tackling the expression 1/9 x^(2) - 10/3 x^(2 )y + ?, and our mission is to figure out what that question mark should be to complete the perfect square. So, buckle up, grab your thinking caps, and let's get started on this mathematical adventure!

Understanding Perfect Square Trinomials

Before we jump into solving our specific problem, let's make sure we're all on the same page about what a perfect square trinomial actually is. Think of it as a special type of trinomial – a polynomial with three terms – that results from squaring a binomial (an expression with two terms). In simpler terms, it's what you get when you multiply a binomial by itself. For example, (a + b)^(2) is a binomial squared. When you expand it, you get a^(2) + 2ab + b^(2), which is a perfect square trinomial. Similarly, (a - b)^(2) expands to a^(2) - 2ab + b^(2), another perfect square trinomial.

The key here is recognizing the pattern. A perfect square trinomial always follows one of these two forms:

  • a^(2) + 2ab + b^(2) = (a + b)^(2)
  • a^(2) - 2ab + b^(2) = (a - b)^(2)

Notice how the first and last terms (a^(2) and b^(2)) are perfect squares, and the middle term (2ab) is twice the product of the square roots of the first and last terms. This pattern is our secret weapon for solving our puzzle! We can use it to identify the missing term in our expression and complete the perfect square. By understanding this fundamental concept, we lay the groundwork for confidently tackling the problem at hand and similar challenges in the future. So, let's keep this pattern in mind as we move forward and apply it to our specific question. Recognizing these patterns is crucial not just for solving this particular problem, but for building a solid foundation in algebra and beyond. It's like learning a new language – once you grasp the grammar, you can start to understand and construct more complex sentences. In this case, the grammar is the structure of perfect square trinomials, and the sentences are the algebraic expressions we'll be working with. So, let's continue building our mathematical vocabulary and see how this knowledge helps us crack the code!

Deconstructing the Given Expression: 1/9 x^(2) - 10/3 x^(2 )y

Now that we've got a solid grasp of what perfect square trinomials are, let's turn our attention to the expression we need to complete: 1/9 x^(2) - 10/3 x^(2 )y + ?. The first step in solving any math puzzle is to break it down into smaller, manageable pieces. Think of it like examining a complex machine – you need to understand each part individually before you can see how they all fit together. In this case, we'll dissect the given expression, identify its components, and see how they relate to the perfect square trinomial pattern we discussed earlier.

First, let's focus on the terms we do have. We have 1/9 x^(2), which looks like it could be our 'a^(2)' term from the perfect square trinomial pattern. Then we have -10/3 x^(2 )y, which seems like it might be the '-2ab' term. And of course, we have the question mark, representing the missing 'b^(2)' term that we need to find. The next step is to try and match these terms to the perfect square trinomial pattern. Remember, the pattern is a^(2) - 2ab + b^(2) (since our middle term is negative). So, let's see if we can figure out what 'a' and 'b' are in our specific expression.

Looking at 1/9 x^(2), we can see that it's a square. What's the square root of 1/9 x^(2)? Well, the square root of 1/9 is 1/3, and the square root of x^(2) is x. So, it seems like 'a' could be 1/3 x. Now let's move on to the middle term, -10/3 x^(2 )y. This term is supposed to be '-2ab'. We already have a candidate for 'a' (1/3 x), so let's plug that in and see if we can solve for 'b'. We have -10/3 x^(2 )y = -2 * (1/3 x) * b. Simplifying this equation will help us isolate 'b' and figure out its value. Once we know 'b', we can easily find 'b^(2)', which is the missing term we're after. This process of deconstruction is crucial for problem-solving in mathematics. By breaking down a complex problem into smaller, more manageable parts, we can apply our knowledge and tools more effectively. It's like building a house – you start with the foundation, then the walls, then the roof. Each step builds upon the previous one, and eventually, you have a complete structure. Similarly, in mathematics, each step in our analysis brings us closer to the final solution. So, let's continue our detective work and see if we can uncover the value of 'b'!

Cracking the Code: Solving for the Missing Term

Alright, let's get down to the nitty-gritty and actually solve for that missing term! Remember, we've identified that 1/9 x^(2) corresponds to a^(2) and -10/3 x^(2 )y corresponds to -2ab in our perfect square trinomial pattern. We've also deduced that 'a' is likely 1/3 x. Now, our mission is to find 'b' and then calculate b^(2) to fill in the blank.

To find 'b', we'll use the equation -10/3 x^(2 )y = -2 * (1/3 x) * b. This equation is the key to unlocking the mystery. It connects the middle term of our expression to the 'a' and 'b' components of the perfect square trinomial. To solve for 'b', we need to isolate it on one side of the equation. We can do this by performing algebraic manipulations – the same kind of moves you'd use in any equation-solving scenario.

First, let's simplify the right side of the equation. -2 multiplied by 1/3 x gives us -2/3 x. So, our equation now looks like this: -10/3 x^(2 )y = -2/3 x * b. Now, to get 'b' by itself, we need to divide both sides of the equation by -2/3 x. Remember, dividing by a fraction is the same as multiplying by its reciprocal. So, we'll multiply both sides by -3/2x. On the left side, (-10/3 x^(2 )y) * (-3/2x) simplifies to 5xy. On the right side, (-2/3 x * b) * (-3/2x) simplifies to b. So, we've found that b = 5xy!

But we're not quite done yet. Remember, we need to find the missing term, which is b^(2). So, we need to square our value of 'b'. (5xy)^(2) is 5xy multiplied by itself, which gives us 25x(2)y(2). And there you have it! We've cracked the code and found the missing term. The value that completes the perfect square trinomial is 25x(2)y(2). This process of solving for the missing term highlights the power of algebraic manipulation. By carefully applying the rules of algebra, we can unravel complex equations and extract the information we need. It's like being a detective, using clues and logic to solve a mystery. And in this case, the mystery was the missing term in our perfect square trinomial. So, now that we've found it, let's put it all together and see the complete picture!

The Grand Finale: Completing the Perfect Square Trinomial

Drumroll, please! We've done the detective work, solved the equations, and now it's time for the grand reveal. We're going to put all the pieces together and show the complete perfect square trinomial. Remember, our original expression was 1/9 x^(2) - 10/3 x^(2 )y + ?, and we've determined that the missing term is 25x(2)y(2).

So, the completed perfect square trinomial is: 1/9 x^(2) - 10/3 x^(2 )y + 25x(2)y(2). But we're not going to stop there! It's always a good idea to double-check our work, and in this case, we can do that by factoring the trinomial back into its binomial squared form. This will confirm that we've indeed created a perfect square trinomial.

Recall that the perfect square trinomial pattern is a^(2) - 2ab + b^(2) = (a - b)^(2). We identified earlier that 'a' is 1/3 x and 'b' is 5xy. So, let's plug those values into the binomial squared form: (1/3 x - 5xy)^(2). Now, let's expand this binomial to see if it matches our completed trinomial. (1/3 x - 5xy)^(2) means (1/3 x - 5xy) * (1/3 x - 5xy). Using the FOIL method (First, Outer, Inner, Last) to expand this, we get:

  • First: (1/3 x) * (1/3 x) = 1/9 x^(2)
  • Outer: (1/3 x) * (-5xy) = -5/3 x^(2)y
  • Inner: (-5xy) * (1/3 x) = -5/3 x^(2)y
  • Last: (-5xy) * (-5xy) = 25x(2)y(2)

Combining these terms, we get 1/9 x^(2) - 5/3 x^(2)y - 5/3 x^(2)y + 25x(2)y(2), which simplifies to 1/9 x^(2) - 10/3 x^(2 )y + 25x(2)y(2). And guess what? It matches our completed trinomial! This confirms that we've correctly identified the missing term and created a perfect square trinomial. This final step of factoring and expanding is a powerful way to verify our solution. It's like having a second opinion on your work. By going through the process in reverse, we can be confident that we've arrived at the correct answer. So, congratulations, math detectives! We've successfully solved the puzzle and completed the perfect square trinomial. This is a testament to the power of understanding patterns and applying algebraic techniques. Keep practicing, and you'll be a master of perfect square trinomials in no time!

Real-World Applications and Further Exploration

So, we've successfully navigated the world of perfect square trinomials and found our missing term. But you might be wondering,