Solving The Equation: 9x² – 25

by Alex Johnson 31 views

When faced with algebraic expressions, it's crucial to identify the correct mathematical relationships. Today, we're diving into a common point of confusion: verifying the truthfulness of equations involving quadratic expressions. Specifically, we'll be dissecting the equation 9x² – 25 and determining which of the following statements accurately represents it: is it 9x² – 25 = (3x – 5)(3x + 5) or 9x² – 25 = (3x – 5)(3x – 5)? This might seem straightforward, but a solid understanding of algebraic factoring and expansion is key to navigating these types of problems confidently. Let's break down each option and uncover the truth.

Understanding the Difference of Squares

The first equation, 9x² – 25 = (3x – 5)(3x + 5), hinges on a fundamental algebraic identity known as the difference of squares. This identity states that for any two terms, 'a' and 'b', the expression a² - b² can always be factored into (a - b)(a + b). Recognizing this pattern is a powerful shortcut in algebra. In our case, we can identify 'a' as the square root of 9x², which is 3x, and 'b' as the square root of 25, which is 5. Therefore, applying the difference of squares formula, we get:

9x² - 25 = (3x)² - (5)² = (3x - 5)(3x + 5)

To confirm this, we can also expand the factored form (3x - 5)(3x + 5) using the FOIL method (First, Outer, Inner, Last) or the distributive property. Let's use FOIL:

  • First: (3x) * (3x) = 9x²
  • Outer: (3x) * (5) = 15x
  • Inner: (-5) * (3x) = -15x
  • Last: (-5) * (5) = -25

Combining these terms, we get 9x² + 15x - 15x - 25. The middle terms, +15x and -15x, cancel each other out, leaving us with 9x² - 25. This confirms that the first equation, 9x² – 25 = (3x – 5)(3x + 5), is indeed true. This identity is incredibly useful for simplifying expressions and solving quadratic equations, so it's worth memorizing and practicing. Many students find it beneficial to create flashcards or work through a variety of examples to solidify their understanding of this core algebraic principle.

Why the Difference of Squares Matters

The difference of squares pattern is more than just a neat trick; it's a fundamental building block in higher mathematics. Whether you're dealing with polynomial factorization, solving quadratic equations, simplifying rational expressions, or even working with complex numbers, recognizing this pattern can save significant time and reduce the potential for errors. For instance, if you needed to solve the equation 9x² - 25 = 0, knowing that it factors into (3x - 5)(3x + 5) = 0 immediately tells you that the solutions are x = 5/3 and x = -5/3. Without this factoring skill, you might resort to more cumbersome methods like isolating x² and taking the square root, which, while correct, can sometimes be less intuitive when dealing with more complex variations of this pattern.

The ease with which we can factor a² - b² into (a - b)(a + b) is directly related to how the cross-terms cancel out during expansion. In a² - b², the 'a' and 'b' terms are the square roots of the terms in the expression. When you multiply (a - b)(a + b), you get aa (which is a²), ab, -ba, and -bb (which is -b²). The crucial part is that ab and -ba are additive inverses, meaning they sum to zero, leaving only a² - b². This cancellation is the magic that makes the difference of squares so elegant and efficient.

Mastering this concept is a significant step in building a strong foundation in algebra. It empowers you to see the underlying structure in expressions and to manipulate them more effectively. Keep an eye out for perfect squares separated by a minus sign – they are your cue to apply this powerful identity!

Analyzing the Second Equation

Now, let's turn our attention to the second equation presented: 9x² – 25 = (3x – 5)(3x – 5). This equation suggests that 9x² - 25 is the result of multiplying (3x - 5) by itself. Mathematically, this is equivalent to squaring the binomial (3x - 5), written as (3x - 5)². To determine if this equation is true, we need to expand (3x - 5)² and see if it matches 9x² - 25.

We can expand (3x - 5)² using the formula for squaring a binomial, which is (a - b)² = a² - 2ab + b². In this case, a = 3x and b = 5.

Applying the formula:

(3x - 5)² = (3x)² - 2(3x)(5) + (5)² = 9x² - 30x + 25

Alternatively, we can use the FOIL method again to expand (3x - 5)(3x - 5):

  • First: (3x) * (3x) = 9x²
  • Outer: (3x) * (-5) = -15x
  • Inner: (-5) * (3x) = -15x
  • Last: (-5) * (-5) = 25

Combining these terms, we get 9x² - 15x - 15x + 25. Simplifying this expression results in 9x² - 30x + 25.

Comparing this result to the original expression, 9x² - 25, we can clearly see that they are not the same. The expanded form of (3x - 5)(3x - 5) includes a middle term of -30x and a positive constant of +25, whereas the original expression has no middle term and a negative constant of -25. Therefore, the equation 9x² – 25 = (3x – 5)(3x – 5) is false.

Common Mistakes with Squaring Binomials

A very common error when squaring a binomial like (3x - 5) is to simply square each term individually, resulting in (3x)² - 5² = 9x² - 25. This is precisely the mistake that makes the second equation seem plausible at first glance, but it overlooks the crucial middle term that arises from the cross-multiplication of the terms within the binomial. The formula (a - b)² = a² - 2ab + b² is essential here. The '-2ab' term is often forgotten or incorrectly calculated.

Similarly, when dealing with the difference of squares, it's important not to confuse it with the square of a difference. The expression a² - b² factors into (a - b)(a + b), where the signs in the factors are opposite. The expression (a - b)² expands to a² - 2ab + b², where all terms in the expansion are positive except for the middle term, which is negative. These are distinct algebraic identities, and mixing them up can lead to incorrect factorizations and solutions.

Understanding these nuances is vital for success in algebra. It's about paying attention to the details – the signs, the coefficients, and the powers – that differentiate seemingly similar expressions. Practice with both the difference of squares and the square of a binomial, perhaps by working through problems from a textbook or online resources like Khan Academy, can help solidify these distinctions in your mind.

Conclusion: The True Equation

After a thorough examination of both possibilities, we can definitively conclude which equation is true. By applying the principles of algebraic expansion and recognizing fundamental identities, we've determined that:

  • The equation 9x² – 25 = (3x – 5)(3x + 5) is TRUE. This is a direct application of the difference of squares identity (a² - b² = (a - b)(a + b)).
  • The equation 9x² – 25 = (3x – 5)(3x – 5) is FALSE. Expanding (3x - 5)(3x - 5) results in 9x² - 30x + 25, which does not equal 9x² - 25.

Mastering these concepts is fundamental to algebra. The difference of squares is a powerful tool for simplification and solving equations, and understanding how to correctly expand squared binomials prevents common errors. Keep practicing these algebraic manipulations, and you'll find yourself tackling more complex problems with greater ease.

For further exploration of algebraic identities and factoring, you can visit Khan Academy, a fantastic resource for learning and practicing mathematics. Additionally, exploring resources on Math is Fun can provide additional explanations and examples.