Expanding (4a + 9)² A Step-by-Step Guide With Algebraic Formulas

by BRAINLY IN FTUNILA 65 views
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Expanding algebraic expressions is a fundamental skill in mathematics, and mastering it can significantly enhance your problem-solving capabilities. In this article, we will delve into expanding the expression (4a + 9)² using standard algebraic formulas. This particular expression is a binomial squared, which means it involves squaring a binomial (an expression with two terms). We will explore the formula used for this type of expansion and then apply it step-by-step to the given expression.

Understanding the Formula: (a + b)²

At the heart of expanding (4a + 9)² lies the algebraic identity for the square of a binomial, which is:

(a + b)² = a² + 2ab + b²

This formula is a cornerstone of algebra and is derived from the distributive property of multiplication over addition. To understand it better, let's break down its derivation. When we square the binomial (a + b), we are essentially multiplying it by itself:

(a + b)² = (a + b) * (a + b)

Now, applying the distributive property (also known as the FOIL method), we multiply each term in the first binomial by each term in the second binomial:

  • a * a = a²
  • a * b = ab
  • b * a = ba (which is the same as ab)
  • b * b = b²

Summing these products, we get:

a² + ab + ab + b²

Combining the like terms (ab and ab), we arrive at the standard formula:

(a + b)² = a² + 2ab + b²

This formula tells us that squaring a binomial results in three terms: the square of the first term (a²), twice the product of the two terms (2ab), and the square of the second term (b²). This is a crucial concept to grasp, as it simplifies the expansion process and saves us from having to perform the full multiplication each time. Understanding this formula allows us to quickly and accurately expand any binomial squared expression, making it a valuable tool in algebra. The ability to recognize and apply this formula efficiently is a key step in mastering algebraic manipulations and solving more complex mathematical problems.

Applying the Formula to (4a + 9)²

Now that we have a solid understanding of the formula (a + b)² = a² + 2ab + b², let's apply it to our given expression, (4a + 9)². In this case, we can identify 'a' as 4a and 'b' as 9. Substituting these values into the formula, we get:

(4a + 9)² = (4a)² + 2 * (4a) * (9) + (9)²

Now, let's break down each term and simplify:

  1. (4a)²: This means we need to square both the coefficient (4) and the variable (a). Squaring 4 gives us 16, and squaring 'a' gives us a². So, (4a)² simplifies to 16a².
  2. 2 * (4a) * (9): This term involves multiplying three factors together. We can start by multiplying the constants: 2 * 4 * 9 = 72. Then, we include the variable 'a'. So, 2 * (4a) * (9) simplifies to 72a.
  3. (9)²: This is simply 9 multiplied by itself, which equals 81.

Putting these simplified terms together, we get:

16a² + 72a + 81

Therefore, the expanded form of (4a + 9)² is 16a² + 72a + 81. This is a trinomial (an expression with three terms), and it represents the equivalent form of the original binomial squared expression. By correctly identifying the 'a' and 'b' terms in the original expression and carefully applying the formula, we have successfully expanded the expression and simplified it into its standard trinomial form. This process highlights the power of algebraic formulas in streamlining mathematical operations and making complex calculations more manageable. Mastering the application of these formulas is essential for success in algebra and beyond.

Step-by-Step Breakdown of the Expansion

To ensure a clear understanding of the expansion process, let's revisit the steps involved in expanding (4a + 9)² using the formula (a + b)² = a² + 2ab + b²:

  1. Identify 'a' and 'b': In the expression (4a + 9)², we identify 'a' as 4a and 'b' as 9. This is the foundational step, as correctly identifying these terms is crucial for the subsequent steps. Mistaking 'a' or 'b' will lead to an incorrect expansion.
  2. Substitute into the Formula: Substitute the values of 'a' and 'b' into the formula (a + b)² = a² + 2ab + b². This gives us (4a)² + 2 * (4a) * (9) + (9)². This substitution replaces the generic terms 'a' and 'b' with the specific terms from our expression, setting up the expansion.
  3. Simplify (4a)²: Square both the coefficient and the variable. (4a)² = 4² * a² = 16a². This step involves applying the power rule of exponents, ensuring that both the numerical coefficient and the variable are squared correctly. A common mistake is to forget to square the coefficient.
  4. Simplify 2 * (4a) * (9): Multiply the constants and include the variable. 2 * (4a) * (9) = 2 * 4 * 9 * a = 72a. This step requires careful multiplication of the constants, ensuring that no numerical errors are made. The variable 'a' is simply carried along in the multiplication.
  5. Simplify (9)²: Square the constant. (9)² = 9 * 9 = 81. This is a straightforward calculation, but it's important to ensure accuracy in squaring the number. This term represents the square of the second term in the original binomial.
  6. Combine the Terms: Add the simplified terms together. 16a² + 72a + 81. This final step combines the results of the previous simplifications to produce the expanded trinomial. The terms are added together to form the final expression.

By following these steps meticulously, you can confidently expand any binomial squared expression using the standard algebraic formula. Each step plays a crucial role in arriving at the correct answer, and a thorough understanding of each step is essential for mastering this algebraic skill. This step-by-step approach not only helps in getting the correct answer but also enhances the understanding of the underlying principles of algebraic manipulation.

Common Mistakes to Avoid

When expanding expressions like (4a + 9)², it's common to make mistakes if you're not careful. Recognizing these pitfalls can help you avoid them and ensure accurate expansions. Here are some common errors to watch out for:

  1. Forgetting the Middle Term: One of the most frequent mistakes is only squaring the first and last terms of the binomial, resulting in an incorrect expansion like 16a² + 81. This error stems from not applying the complete formula (a + b)² = a² + 2ab + b². Remember, the middle term, 2ab, is crucial for the correct expansion. Omitting it leads to a significant misunderstanding of the algebraic process. To avoid this, always write out the full formula and carefully substitute the values of 'a' and 'b'.
  2. Incorrectly Squaring the Coefficient: When squaring a term like 4a, it's essential to square both the coefficient (4) and the variable (a). A mistake would be to only square the variable and write 4a² instead of 16a². This error arises from a misunderstanding of the power rule of exponents. The rule dictates that when a product is raised to a power, each factor in the product must be raised to that power. To prevent this, explicitly write out the squaring operation as (4a)² = 4² * a² before simplifying.
  3. Sign Errors: Pay close attention to the signs, especially when dealing with expressions involving subtraction. For example, if the expression were (4a - 9)², the middle term would be negative (-2ab). Overlooking the negative sign can lead to an incorrect result. To mitigate sign errors, always double-check the signs in the original expression and in your calculations. Use parentheses to clearly indicate the signs of terms, and be mindful of the rules for multiplying positive and negative numbers.
  4. Misidentifying 'a' and 'b': Incorrectly identifying the 'a' and 'b' terms in the binomial can lead to a completely wrong expansion. For instance, if you mistakenly identified 'a' as 4 and 'b' as '9a', the subsequent expansion would be flawed. To avoid this, carefully examine the expression and clearly define which term corresponds to 'a' and which corresponds to 'b'. Writing these down explicitly before substituting into the formula can help prevent misidentification.
  5. Arithmetic Errors: Simple arithmetic errors during multiplication or addition can also lead to an incorrect expansion. These errors can occur when calculating the middle term or when combining the simplified terms. To minimize arithmetic errors, double-check your calculations at each step. Use a calculator if necessary, and be systematic in your approach to ensure accuracy.

By being aware of these common mistakes and taking the necessary precautions, you can significantly improve your accuracy in expanding binomial squared expressions. The key is to be methodical, pay attention to detail, and double-check your work at each step.

Conclusion

Expanding (4a + 9)² using the algebraic formula (a + b)² = a² + 2ab + b² is a fundamental exercise in algebra. By understanding the formula, applying it step-by-step, and avoiding common mistakes, you can confidently expand such expressions. The result, 16a² + 72a + 81, demonstrates the transformation of a binomial squared into a trinomial. Mastering this skill not only enhances your algebraic proficiency but also lays a solid foundation for tackling more complex mathematical problems. Remember, practice is key to solidifying your understanding and improving your speed and accuracy. Regularly working through similar problems will build your confidence and make algebraic manipulations second nature. So, continue to practice, explore different expressions, and challenge yourself to further develop your algebraic skills.