Evaluate A² − 3b When A = 2 And B = -3

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Evaluate a² − 3b When a = 2 and b = -3

Let's dive into evaluating the algebraic expression a² − 3b when a equals 2 and b equals -3. This kind of problem is fundamental in algebra and helps solidify understanding of variable substitution and order of operations. It’s a straightforward process, but accuracy is key to arriving at the correct answer. Stick with me, and we’ll break it down step by step. So, grab your pencils, and let's get started!

Understanding the Expression

Before we jump into plugging in the values, let's make sure we understand what the expression a² − 3b actually means. The expression contains two variables, a and b, and it involves exponentiation, multiplication, and subtraction. The term a² means a raised to the power of 2, or a multiplied by itself. The term 3b means 3 multiplied by b. The entire expression tells us to first square the value of a, then multiply 3 by the value of b, and finally subtract the second result from the first. Understanding this structure is crucial because it dictates the order in which we perform the operations. Remember folks, adhering to the correct order of operations is essential for accurately evaluating any mathematical expression. So, now that we're all on the same page, let's proceed with substituting the given values into our expression.

Substituting the Values

Now comes the fun part: substituting the given values into the expression. We are given that a = 2 and b = -3. This means we replace every instance of a in the expression with 2, and every instance of b with -3. When we substitute these values into a² − 3b, we get (2)² − 3(-3). See how we've simply replaced the variables with their corresponding numerical values? Make sure you pay close attention to signs, especially when dealing with negative numbers. A small mistake with the signs can lead to a completely different result. Remember, mathematics is all about precision, so let's be meticulous as we move forward. Now that we have successfully substituted the values, we can proceed to simplify the expression according to the order of operations.

Applying the Order of Operations

To correctly evaluate the expression (2)² − 3(-3), we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our expression, we first deal with the exponent, then the multiplication, and finally the subtraction. Let's start with the exponent. (2)² means 2 raised to the power of 2, which is 2 * 2 = 4. So, we can replace (2)² with 4 in our expression. Now we have 4 − 3(-3). Next, we perform the multiplication. 3(-3) means 3 multiplied by -3, which equals -9. So, we can replace 3(-3) with -9 in our expression. Now we have 4 − (-9). Finally, we perform the subtraction. Subtracting a negative number is the same as adding its positive counterpart. So, 4 − (-9) is the same as 4 + 9, which equals 13. Therefore, the final result of evaluating the expression a² − 3b when a = 2 and b = -3 is 13. Keep practicing, and you'll become a master of order of operations in no time!

Step-by-Step Solution

Let's recap the step-by-step solution to make sure we've got it all crystal clear:

  1. Write down the expression: a² − 3b
  2. Substitute the values: Given a = 2 and b = -3, substitute these values into the expression to get (2)² − 3(-3).
  3. Evaluate the exponent: Calculate (2)², which equals 4. The expression becomes 4 − 3(-3).
  4. Perform the multiplication: Calculate 3(-3), which equals -9. The expression becomes 4 − (-9).
  5. Perform the subtraction: Subtract -9 from 4, which is the same as adding 9 to 4. 4 − (-9) = 4 + 9 = 13.

Therefore, a² − 3b = 13 when a = 2 and b = -3.

Common Mistakes to Avoid

When evaluating algebraic expressions, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct answer. One common mistake is neglecting the order of operations. Remember to follow PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Another frequent error is mishandling negative signs. Be especially careful when substituting negative values and when performing multiplication or subtraction with negative numbers. For example, subtracting a negative number is the same as adding its positive counterpart. Also, make sure to square the correct value. In the expression a², you are squaring the value of a, not multiplying a by 2. Finally, double-check your arithmetic. Simple calculation errors can easily lead to incorrect answers. By being mindful of these common mistakes, you can improve your accuracy and confidence in evaluating algebraic expressions. Keep an eye out for these traps, and you'll be solving expressions like a pro in no time!

Practice Problems

To solidify your understanding of evaluating algebraic expressions, here are a few practice problems for you to try. Remember to follow the order of operations and pay close attention to signs:

  1. Evaluate x² + 2y when x = 3 and y = -2.
  2. Evaluate 2p − q² when p = -1 and q = 4.
  3. Evaluate m² − 4n when m = -2 and n = 1.
  4. Evaluate 3a + b² when a = 5 and b = -3.
  5. Evaluate c² − 2d when c = 4 and d = -1.

Work through these problems carefully, and then check your answers. The more you practice, the more comfortable and confident you will become with evaluating algebraic expressions. So, grab a pencil and paper, and give these problems a shot. Happy solving!

Conclusion

In conclusion, evaluating the expression a² − 3b when a = 2 and b = -3 involves substituting the given values into the expression and then simplifying according to the order of operations. We found that when a = 2 and b = -3, the expression a² − 3b equals 13. This exercise highlights the importance of understanding variable substitution, the order of operations, and careful handling of signs. By mastering these fundamental concepts, you'll be well-equipped to tackle more complex algebraic problems with confidence. Remember to practice regularly and pay attention to detail, and you'll be well on your way to becoming an algebra whiz! Great job working through this problem with me. Keep up the excellent work, and I'll see you in the next mathematical adventure! You got this! Huzzah!