Question

Happy New Year!

plz solve this question

Answer 1

Answer:

816√2

Step-by-step explanation:

Given Equation is x^(1/4) + x^(-1/4) = 2√2

Let x^(1/4) = a.

⇒ a + 1/a = 2√2

⇒ a² + 1 = 2√2a

⇒ a² - 2√2a + 1 = 0

Here, a = 1, b = -2√2, c = 1

∴ D = b² - 4ac

      = (-2√2)² - 4(1)(1)

      = 8 - 4

      = 4.

The solutions are:

⇒ x = -b ± √D/2a

      = -(-2√2) ± √4/2

      = 2√2 ± 2/2

      = 1 + √2, √2 - 1.

(i)

x^(1/4) = 1 + √2

Taking power 4 on both sides, we get

x = (1 + √2)⁴

x = 17 + 12√2

(ii)

x^(1/4) = √2 - 1

x = 17 - 12√2.

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Now,

⇒ x² = (17 + 12√2)²

        = 577 + 408√2.

Then,

(1/x²) = 1/(577 + 408√2)

        = 1/(577 + 408√2) * (577 - 408√2)/(577 - 408√2)

        = (577 - 408√2)/(577)² - (408√2)²

        = 577 - 408√2/(1)

        = 577 - 408√2

Hence:

⇒ x² - 1/x²

⇒ (577 + 408√2) - (577 - 408√2)

⇒ 577 + 408√2 - 577 + 408√2

⇒ 816√2.

Note: You can consider x² as any value, the result will be the same.

---------------------------------- Happy New year -----------------------------