Which choice is equivalent to the fraction below when X is an appropriate value? Hint: Rationalize denominator and simplify

Option A. –2 – 2√2

Step-by-step explanation:

√8 / √2 – 2

The above expression can be simplified as follow.

√8 / (√2 – 2)

Rationalise by multiplying the numerator and denominator with the conjugate of (√2 – 2) i.e (√2 + 2). This is illustrated below:

√8 / (√2 – 2) × (√2 + 2)/(√2 + 2)

√8(√2 + 2) / (√2 – 2)(√2 + 2)

√16 + 2√8 / 2 + 2√2 – 2√2 – 4

√16 + 2√8 / 2 – 4

√16 + 2√8 / –2

Recall

√16 = 4

√8 = √(4×2) = √4 × √2 = 2√2

Thus,

√16 + 2√8 / –2 = 4 + 2×2√2 / –2

= (4 + 4√2) / –2

= 4/–2 + 4√2/–2

= –2 – 2√2

Therefore,

√8 / (√2 – 2) = –2 – 2√2

Thus, option A gives the correct answer to the question.