For which value of x will the expression 11.98 × 11.98 + 11.98 × x + 0.02 × 0.02 represent a perfect square?

Introduction / Context:
This problem tests your recognition of algebraic identities in decimal form, specifically the expansion of a perfect square. Recognizing patterns like (a + b)^2 and matching them to given expressions is a powerful technique for simplifying algebra and for quickly identifying missing coefficients.

Given Data / Assumptions:

• The expression is 11.98 × 11.98 + 11.98 × x + 0.02 × 0.02.
• We are told that for some value of x this expression becomes a perfect square.
• The options provide possible values for x: 0.01, 0.02, 0.03, 0.04, and 0.05.
• We must determine the value of x that makes the entire expression equal to (11.98 + 0.02)^2 or a similar pattern.

Concept / Approach:
Recall the identity (a + b)^2 = a^2 + 2ab + b^2. In the given expression, 11.98 × 11.98 plays the role of a^2, 0.02 × 0.02 plays the role of b^2, and 11.98 × x should correspond to the middle term 2ab. We match these pieces and solve for x by equating 11.98 × x with 2 × 11.98 × 0.02.

Step-by-Step Solution:
Identify a and b: a = 11.98 and b = 0.02.A perfect square of the form (a + b)^2 expands to a^2 + 2ab + b^2.In our expression, a^2 = 11.98 × 11.98 and b^2 = 0.02 × 0.02, so those parts already match.The middle term in the perfect square would be 2ab = 2 × 11.98 × 0.02.Compute 2 × 11.98 × 0.02 = 11.98 × 0.04.In the given expression, the middle term is 11.98 × x.For the expressions to match, we must have 11.98 × x = 11.98 × 0.04.Therefore, x = 0.04.

Verification / Alternative check:
Substitute x = 0.04 back into the expression. It becomes 11.98 × 11.98 + 11.98 × 0.04 + 0.02 × 0.02. This matches the expansion of (11.98 + 0.02)^2 exactly, which is 12.00^2. Therefore, the expression becomes 144, a perfect square. No other value of x will produce this exact algebraic identity, so 0.04 is the unique correct answer among the choices.

Why Other Options Are Wrong:
Any other value for x would make the middle term 11.98 × x different from the required 2ab term. For example, x = 0.02 would give 11.98 × 0.02, which is only half the required value. Similarly, x = 0.03 or 0.05 would give mismatched coefficients, meaning the expression would no longer be a perfect square and would not simplify to (a + b)^2 for any a and b.

Common Pitfalls:
One common mistake is to confuse the structure of the identity and think that the middle term should be a × b instead of 2ab. This leads to an incorrect factor of 2 in the value of x. Another error is to try plugging in each option directly without recognizing the identity, which can be time consuming and prone to arithmetic slips. Spotting the pattern a^2 + 2ab + b^2 and using it directly is the fastest and most reliable method.

Final Answer:
The expression becomes a perfect square when x is 0.04.