Question

y = 2x² + 7x - 3
Quantity A: x
Quantity B: y

• A. Quantity A is greater.
• B. Quantity B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.
• E.

Hint:

For algebraic comparison problems, testing a few strategic numbers is a powerful technique. Always consider testing zero, a positive integer (like 1 or 2), and a negative integer (like -1). These three values often reveal whether the relationship is constant or variable.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are given a functional relationship between two variables, x and y, in the form of a quadratic equation. We need to compare the value of the input, x, with the value of the output, y. The relationship is not fixed and will change depending on the value of x.
Step 2: Key Approach:
The most effective method is to test different values for x and calculate the corresponding value of y. If we can find one value of x where x>y and another value where y>x, then the relationship cannot be determined. It's good practice to test simple integers like 0, 1, and -1.
Step 3: Detailed Explanation:
Let's test a few different values for x.
Case 1: Let x = 0.
Substitute x = 0 into the given equation:
\[ y = 2(0)^2 + 7(0) - 3 \] \[ y = 0 + 0 - 3 \] \[ y = -3 \] In this case:
- Quantity A = x = 0
- Quantity B = y = -3
Comparing them, we find that $0>-3$. So, for x=0, Quantity A is greater.
Case 2: Let x = 1.
Substitute x = 1 into the given equation:
\[ y = 2(1)^2 + 7(1) - 3 \] \[ y = 2(1) + 7 - 3 \] \[ y = 2 + 7 - 3 = 6 \] In this case:
- Quantity A = x = 1
- Quantity B = y = 6
Comparing them, we find that $6>1$. So, for x=1, Quantity B is greater.
Step 4: Final Answer:
In our first test case (x=0), we found that Quantity A was greater than Quantity B. In our second test case (x=1), we found that Quantity B was greater than Quantity A. Since the relationship between x and y changes depending on the value of x, a consistent comparison is not possible. Therefore, the relationship cannot be determined from the information given.