Correct Answer - Option 3 : √6 + √7

**Given:**

(x^{4}) + (1/x^{4}) = 674

**Formula Used:**

(x^{4}) + (1/x^{4}) = k, then (x^{2}) + (1/x^{2}) = √(k + 2)

(x^{2}) + (1/x^{2}) = m, then (x) + (1/x) = √(m + 2)

(x^{2}) + (1/x^{2}) = m, then (x) - (1/x) = √(m - 2)

**Calculation:**

(x^{4}) + (1/x^{4}) = 674

Then,

(x^{2}) + (1/x^{2}) = √(674 + 2)

⇒ (x^{2}) + (1/x^{2}) = √676

⇒ (x^{2}) + (1/x^{2}) = 26

Now,

(x) + (1/x) = √(26 + 2)

⇒ (x) + (1/x) = √28

⇒ (x) + (1/x) = 2√7 ...(i)

And,

(x) - (1/x) = √(26 - 2)

⇒ (x) - (1/x) = √24

⇒ (x) - (1/x) = 2√6 ...(ii)

Adding equation i and ii

[(x) + (1/x)] + [(x) - (1/x)] = 2√7 + 2√6

⇒ 2x = 2(√7 + √6)

⇒ x = (√7 + √6)

**∴**** The value of x is (√7 + √6)**