If (3 - 4$\sqrt{2}$)(1 + 3$\sqrt{2}$) = a + b$\sqrt{2}$,  find the value of b

Analyze the question: We need to expand the product of two surds and write it in the form a + b√2, then read off b (the coefficient of √2).

Concept: Expand using FOIL (First, Outer, Inner, Last), remembering that √2 × √2 = 2.

Solution:

(3 - 4√2)(1 + 3√2)

• First: 3 × 1 = 3
• Outer: 3 × 3√2 = 9√2
• Inner: -4√2 × 1 = -4√2
• Last: -4√2 × 3√2 = -12(√2 × √2) = -12 × 2 = -24

Now combine:

• Rational part: 3 - 24 = -21, so a = -21
• Surd part: 9√2 - 4√2 = 5√2, so b = 5

So the expression equals -21 + 5√2, which means b = 5.

Check against the options: The value -21 (option C) is a, not b, and that is a common trap.

Tip: In questions like this, always find both a and b and then check which one the question asks for. Examiners often put the value of a among the options as a distractor. Also remember that √2 × √2 = 2, so the "Last" term becomes rational.