The assessment for this lesson is a discussion of a mathematical statement. You will have to determine if the statement is true or false. If it is false, you will explain your reasoning. You will also be asked to respond to at least two of your classmates. Before engaging in the discussion, view the Discussion Guidelines and the Discussion Rubric to ensure that you understand the expectations for this activity. For all real numbers a and b, 2a • b = a2 + b2

only if a-b=0

since (a-b)^2 = a^2-2ab+b^2
so, 2ab = a^2+b^2

Well, according to my mathematical abilities (which are quite colorful, I must say), this statement is false. 2a • b does not equal a² + b² for all real numbers a and b. Let me break it down for you.

Let's take a simple example where a = 2 and b = -3. If we substitute these values into the equation, we get:

2a • b = a² + b²
2(2) • (-3) = (2)² + (-3)²
-12 = 4 + 9
-12 ≠ 13

As you can see, -12 is not equal to 13. Therefore, the statement is false.

Now, if you'll excuse me, I'm off to make balloon animals for the mathletes. Ta-ta!

To determine if the mathematical statement "For all real numbers a and b, 2a • b = a2 + b2" is true or false, we can use a counterexample.

Let's choose two real numbers, a = 2 and b = 3, and substitute them into the equation:

2(2) • 3 = 2^2 + 3^2
6 ≠ 4 + 9
6 ≠ 13

Since the equation does not hold true for the selected values of a = 2 and b = 3, we have found a counterexample.

Therefore, the statement "For all real numbers a and b, 2a • b = a^2 + b^2" is false.

Now, let's respond to at least two classmates:

Classmate 1: I agree with your explanation that the statement is false. It is clear that the equation does not hold true for all real numbers a and b. Good job!

Classmate 2: I see that you found a counterexample for the equation. However, I think there might be a mistake in your calculation. Let me check it.

Let's use the same values, a = 2 and b = 3, and substitute them into the equation:

2(2) • 3 = 2^2 + 3^2
12 = 4 + 9
12 = 13

Oops! It seems I made a mistake in calculating the left side of the equation. I apologize for the confusion. Your initial explanation is correct, and the statement is indeed false.

Remember, always double-check your calculations to avoid errors.

To determine if the mathematical statement is true or false, we need to evaluate it by substituting some real numbers for the variables a and b. Let's start by replacing a with 3 and b with 4 in the equation:

2(3) • 4 = 3^2 + 4^2

Simplifying the left side:

6 • 4 = 9 + 16

24 = 25

Since 24 does not equal 25, the equation is not true. Therefore, the statement is false.

Now, let's explain the reasoning behind this. The equation 2a • b = a^2 + b^2 does not hold true for all real numbers a and b. It might hold true for some specific values of a and b, but in general, it is incorrect.

In this case, we substituted a = 3 and b = 4, which led us to a false statement. This shows that the equation is not universally valid, and there are counterexamples, such as the numbers we just used, where the equation does not hold true.

It is important to note that when determining the truth or falsity of a mathematical statement, we cannot rely solely on one example. In this case, we only substituted one set of values, and it was sufficient to demonstrate that the statement is false. However, it is always advisable to test multiple values to ensure the statement is indeed false or true.

Now let's move on to responding to at least two classmates' discussions.