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

Introduction / Context:
This problem examines your ability to recognise patterns in algebraic expressions, especially perfect square trinomials. A perfect square trinomial can be written as (a + b)^2 or (a - b)^2. Here, decimals are involved, but the same pattern recognition applies directly.

Given Data / Assumptions:

• Expression: 11.98 × 11.98 + 11.98 × x + 0.02 × 0.02.
• We must find x such that the entire expression is a perfect square of a binomial.
• All multiplications are ordinary real-number multiplications.

Concept / Approach:
Recall the identity (a + b)^2 = a^2 + 2ab + b^2. To match our expression with this form, we can let a = 11.98 and b = 0.02. Then a^2 corresponds to 11.98 × 11.98 and b^2 corresponds exactly to 0.02 × 0.02. The middle term 11.98 × x must match 2ab = 2 × 11.98 × 0.02. This condition allows us to solve for x.

Step-by-Step Solution:
Step 1: Set a = 11.98 and b = 0.02. Step 2: Identify a^2 = 11.98 × 11.98, which is present in the expression. Step 3: Identify b^2 = 0.02 × 0.02, which is also present. Step 4: For a perfect square, the middle term must be 2ab = 2 × 11.98 × 0.02. Step 5: Compute 2ab: 2 × 11.98 × 0.02 = 11.98 × 0.04. Step 6: The given middle term is 11.98 × x, so we require 11.98 × x = 11.98 × 0.04. Step 7: Cancel 11.98 (nonzero) from both sides to get x = 0.04.

Verification / Alternative check:
Substitute x = 0.04 into the expression: 11.98 × 11.98 + 11.98 × 0.04 + 0.02 × 0.02 = a^2 + 2ab + b^2, where a = 11.98 and b = 0.02. This equals (11.98 + 0.02)^2 = 12.00^2 = 144, clearly a perfect square. So x = 0.04 works perfectly.

Why Other Options Are Wrong:
0.01, 0.02, and 0.03 all give middle terms 11.98 × x that are smaller than the required 2ab. They do not satisfy the condition 11.98 × x = 11.98 × 0.04.
0.4 makes the middle term too large and completely breaks the perfect square pattern.

Common Pitfalls:
Learners sometimes confuse 2ab with a × b or attempt to match the expression with (a - b)^2 without checking signs. Another common mistake is to be intimidated by decimals. Treating 11.98 and 0.02 symbolically as a and b simplifies the reasoning, and decimals can be handled at the end if necessary.

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