Question

Quantity A: \( x^2 + 1 \)
Quantity B: \( 2x - 1 \)

• 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:

When comparing two algebraic expressions, subtracting one from the other is a powerful technique. If the resulting expression can be shown to be always positive or always negative (e.g., by completing the square to show it's a squared term plus a positive constant), you can determine the relationship definitively.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We are asked to compare two algebraic expressions, one quadratic and one linear. Since no conditions are placed on x, the relationship must hold true for all possible real values of x. A good strategy is to analyze the difference between the two quantities.
Step 2: Key Approach:
Let's analyze the difference: Quantity A - Quantity B.
- If Quantity A - Quantity B>0, then Quantity A is greater.
- If Quantity A - Quantity B<0, then Quantity B is greater.
- If Quantity A - Quantity B = 0, then they are equal.
- If the sign of the difference can change, the relationship cannot be determined.
Step 3: Detailed Explanation:
Let's calculate the difference between Quantity A and Quantity B:
\[ \text{Difference} = (\text{Quantity A}) - (\text{Quantity B}) \] \[ \text{Difference} = (x^2 + 1) - (2x - 1) \] \[ \text{Difference} = x^2 + 1 - 2x + 1 \] \[ \text{Difference} = x^2 - 2x + 2 \] Now, we need to determine if this expression is always positive, always negative, or if its sign can change. We can analyze this quadratic by completing the square.
To complete the square for \(x^2 - 2x\), we take half of the coefficient of x (-2), which is -1, and square it to get 1.
We can rewrite our expression as:
\[ \text{Difference} = (x^2 - 2x + 1) + 1 \] The expression in the parenthesis is a perfect square:
\[ \text{Difference} = (x - 1)^2 + 1 \] Now let's analyze this simplified form.
- The term \( (x - 1)^2 \) is the square of a real number. The square of any real number is always greater than or equal to 0. So, \( (x - 1)^2 \geq 0 \).
- This means the smallest possible value of \( (x - 1)^2 \) is 0 (which occurs when x = 1).
- Therefore, the smallest possible value of the entire expression \( (x - 1)^2 + 1 \) is \( 0 + 1 = 1 \).
So, the difference between Quantity A and Quantity B is always greater than or equal to 1. Since the difference is always positive, Quantity A must always be greater than Quantity B.
Step 4: Final Answer:
The expression \( (x-1)^2 + 1 \) is always positive (in fact, it's always \( \geq 1 \)). This means that \( x^2 + 1>2x - 1 \) for all values of x. Therefore, Quantity A is greater. The correct choice is (A).