Question

If (x⁴) + (1/x⁴) = 674, then find the value of x.

1. 2√7
2. 2√6 + 2√7
3. √6 + √7
4. 2√6
5. 4√6 + √7

Answer 1

Correct Answer - Option 3 : √6 + √7

Given:

(x⁴) + (1/x⁴) = 674 

Formula Used:

(x⁴) + (1/x⁴) = k, then (x²) + (1/x²) = √(k + 2)

(x²) + (1/x²) = m, then (x) + (1/x) = √(m + 2)

(x²) + (1/x²) = m, then (x) - (1/x) = √(m - 2)

Calculation:

(x⁴) + (1/x⁴) = 674

Then,

(x²) + (1/x²) = √(674 + 2)

⇒ (x²) + (1/x²) = √676

⇒ (x²) + (1/x²) = 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)