Expanding And Simplifying Expressions Calculating 13 X 18

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Part a) Expanding and Simplifying (x + 3)(x + 8)

Understanding the Distributive Property

At the heart of expanding and simplifying algebraic expressions lies the distributive property. This property states that for any numbers a, b, and c:

  • a(b + c) = ab + ac

This fundamental rule allows us to multiply a single term by a sum or difference within parentheses. To expand the product of two binomials, such as (x + 3)(x + 8), we apply the distributive property twice, ensuring that each term in the first binomial multiplies each term in the second binomial.

Applying the Distributive Property

To expand (x + 3)(x + 8), we can visualize the process using the acronym FOIL, which stands for:

  • First: Multiply the first terms of each binomial.
  • Outer: Multiply the outer terms of the binomials.
  • Inner: Multiply the inner terms of the binomials.
  • Last: Multiply the last terms of each binomial.

Let's apply the FOIL method to (x + 3)(x + 8):

  1. First: x * x = x²
  2. Outer: x * 8 = 8x
  3. Inner: 3 * x = 3x
  4. Last: 3 * 8 = 24

Combining Like Terms

Now we have expanded the expression:

x² + 8x + 3x + 24

The next step is to simplify by combining like terms. Like terms are those that have the same variable raised to the same power. In this case, 8x and 3x are like terms.

Combining 8x and 3x, we get 11x. Therefore, the simplified expression is:

x² + 11x + 24

Final Result

Thus, the fully expanded and simplified form of (x + 3)(x + 8) is x² + 11x + 24. This quadratic expression represents the product of the two binomials in its simplest form. Understanding how to arrive at this result is crucial for further algebraic manipulations and problem-solving.

Part b) Using the Result to Calculate 13 × 18

Connecting Algebra to Arithmetic

The beauty of algebra lies in its ability to represent numerical relationships in a general form. By substituting specific values for variables, we can apply algebraic results to solve arithmetic problems. In this case, we will leverage the expanded form of (x + 3)(x + 8) to calculate the product of 13 and 18.

Choosing a Suitable Value for x

The key to this approach is to find a value for x that makes (x + 3) equal to 13 or 18 (or a factor of these numbers) and (x + 8) equal to the other number. Let's aim to make (x + 3) equal to 13. This means solving the equation:

x + 3 = 13

Subtracting 3 from both sides, we get:

x = 10

Let's verify if this value of x works for the other binomial. If x = 10, then:

x + 8 = 10 + 8 = 18

Thus, x = 10 is the suitable value that makes (x + 3) equal to 13 and (x + 8) equal to 18.

Substituting x = 10 into the Expanded Form

Now that we have a suitable value for x, we can substitute it into the expanded form of (x + 3)(x + 8), which we found to be x² + 11x + 24:

(10)² + 11(10) + 24

Evaluating the Expression

Now we need to evaluate the numerical expression:

  1. 10² = 100
  2. 11(10) = 110

Adding these values together with 24, we get:

100 + 110 + 24 = 234

The Result

Therefore, using the expanded form and substituting x = 10, we find that 13 × 18 = 234. This demonstrates how algebraic techniques can be employed to simplify numerical calculations. This method can be particularly useful for multiplying numbers that are close to a multiple of 10, as it allows us to leverage the simplicity of squaring and multiplying by 10.

Conclusion: Mastering Algebraic Expansion and Simplification

In this guide, we have explored the process of expanding and simplifying algebraic expressions, focusing on the example of (x + 3)(x + 8). We learned how to apply the distributive property using the FOIL method, combine like terms, and arrive at the simplified expression x² + 11x + 24. Furthermore, we demonstrated how this algebraic result can be used to solve arithmetic problems, specifically calculating 13 × 18. By substituting x = 10 into the expanded form, we efficiently obtained the product 234.

Understanding algebraic expansion and simplification is a crucial skill in mathematics. It forms the foundation for more advanced topics such as factoring, solving equations, and calculus. The ability to manipulate algebraic expressions with confidence opens doors to a deeper understanding of mathematical concepts and their applications in various fields.

This example illustrates the interconnectedness of algebra and arithmetic. By mastering algebraic techniques, we can gain new perspectives on numerical calculations and develop problem-solving strategies that are both efficient and insightful. This knowledge is not only valuable in academic settings but also in everyday life, where mathematical reasoning plays a vital role in decision-making and problem-solving.

This step-by-step guide, complete with examples and explanations, will help you solidify your grasp of these concepts. Remember, practice is key to mastery. The more you practice, the more comfortable and confident you will become in your ability to expand and simplify algebraic expressions. This will not only benefit your academic performance but also enhance your overall mathematical proficiency, enabling you to tackle a wide range of problems with ease and precision.