For which value of x is (11.98 × 11.98 + 11.98 × x + 0.02 × 0.02) a perfect square?

Problem restatement
Find x so the expression matches the identity (a + b)² = a² + 2ab + b² with a = 11.98 and b = 0.02.

Given data

• a = 11.98, b = 0.02
• Target middle term = 2ab = 2 × 11.98 × 0.02 = 0.4792

Concept/Approach
The middle term in the given expression is 11.98 × x, which must equal 0.4792 for a perfect square.

Step-by-Step calculation
11.98 × x = 0.4792x = 0.4792 ÷ 11.98 = 0.04

Verification/Alternative
Substitute x = 0.04: expression becomes 11.98² + 2(11.98)(0.02) + 0.02² = (11.98 + 0.02)² = 12².

Common pitfalls
Using x = 2b (0.04) without checking the coefficient with a (must satisfy 11.98 × x = 2ab).

Final Answer
0.04