Solving 9x^2 - 25: Which Equation Is Correct?
Hey there, math enthusiasts! Today, we're diving into a classic algebraic problem: figuring out which equation correctly represents the expression 9x^2 - 25. This type of problem often pops up in algebra classes and standardized tests, so understanding how to solve it is super valuable. We'll break down each option step by step, making sure you grasp the underlying principles. So, grab your thinking caps, and let’s get started!
The question at hand is: Which of the following equations is true for 9x^2 - 25?
A. 9x^2 - 25 = (3x - 5)(3x - 5) B. 9x^2 - 25 = (3x - 5)(3x + 5) C. 9x^2 - 25 = -(3x + 5)(3x + 5) D. 9x^2 - 25 = -(3x + 5)(3x - 5)
Let's explore each option thoroughly to identify the correct one.
Understanding the Basics: Difference of Squares
Before we jump into the options, let's quickly recap a fundamental concept: the difference of squares. This is a pattern that shows up frequently in algebra, and it's the key to solving our problem. The difference of squares states that:
a^2 - b^2 = (a - b)(a + b)
In simple terms, if you have a squared term minus another squared term, you can factor it into two binomials: one with a subtraction and one with an addition. Recognizing this pattern will make factoring and simplifying expressions much easier.
Now, let's see how this applies to our specific expression, 9x^2 - 25.
Applying the Difference of Squares to 9x^2 - 25
Notice that 9x^2 is a perfect square, as it can be written as (3x)^2. Similarly, 25 is also a perfect square, being 5^2. Therefore, our expression 9x^2 - 25 fits the difference of squares pattern perfectly.
Here, we can identify:
- a = 3x
- b = 5
So, using the difference of squares formula, we can factor 9x^2 - 25 as:
(3x)^2 - (5)^2 = (3x - 5)(3x + 5)
With this factored form in mind, let's evaluate the given options and see which one matches our result.
Evaluating the Options
Now that we've factored the expression using the difference of squares, let's go through each option and see which one holds true.
Option A: 9x^2 - 25 = (3x - 5)(3x - 5)
Option A suggests that 9x^2 - 25 is equal to (3x - 5)(3x - 5). Let's expand the right side to see if it matches the left side.
Expanding (3x - 5)(3x - 5), we get:
(3x - 5)(3x - 5) = (3x)(3x) - (3x)(5) - (5)(3x) + (5)(5) = 9x^2 - 15x - 15x + 25 = 9x^2 - 30x + 25
This result, 9x^2 - 30x + 25, is clearly not equal to 9x^2 - 25. Therefore, Option A is incorrect.
Option B: 9x^2 - 25 = (3x - 5)(3x + 5)
Option B proposes that 9x^2 - 25 is equivalent to (3x - 5)(3x + 5). We've already seen this factorization when we applied the difference of squares formula. Let's expand the right side to confirm.
Expanding (3x - 5)(3x + 5), we have:
(3x - 5)(3x + 5) = (3x)(3x) + (3x)(5) - (5)(3x) - (5)(5) = 9x^2 + 15x - 15x - 25 = 9x^2 - 25
As we can see, the expansion of (3x - 5)(3x + 5) perfectly matches 9x^2 - 25. Thus, Option B is correct.
Option C: 9x^2 - 25 = -(3x + 5)(3x + 5)
Option C suggests that 9x^2 - 25 is equal to the negative of (3x + 5)(3x + 5). Let’s expand the right side and see if it matches.
Expanding -(3x + 5)(3x + 5), we get:
-(3x + 5)(3x + 5) = -[(3x)(3x) + (3x)(5) + (5)(3x) + (5)(5)] = -(9x^2 + 15x + 15x + 25) = -(9x^2 + 30x + 25) = -9x^2 - 30x - 25
This result, -9x^2 - 30x - 25, does not match 9x^2 - 25. Therefore, Option C is incorrect.
Option D: 9x^2 - 25 = -(3x + 5)(3x - 5)
Finally, Option D proposes that 9x^2 - 25 is equivalent to the negative of (3x + 5)(3x - 5). We already know that (3x + 5)(3x - 5) expands to 9x^2 - 25, so let's see what happens when we negate it.
Expanding -(3x + 5)(3x - 5), we get:
-(3x + 5)(3x - 5) = -(9x^2 - 25) = -9x^2 + 25
This result, -9x^2 + 25, does not match 9x^2 - 25. Hence, Option D is also incorrect.
Conclusion: The Correct Equation
After evaluating all the options, we've determined that Option B is the correct one. The equation 9x^2 - 25 = (3x - 5)(3x + 5) accurately represents the factorization using the difference of squares formula.
Understanding and applying the difference of squares pattern is a crucial skill in algebra. It simplifies factoring and solving equations, making complex problems much more manageable. Remember, practice makes perfect, so keep working on these types of problems to solidify your understanding.
If you want to delve deeper into factoring and algebraic identities, a great resource to check out is Khan Academy's Algebra Section. They offer comprehensive lessons and practice exercises to help you master these concepts. Happy solving!
