Question

18. If x + 1 /x= v√5, find the values of x² + 1/x2
and x4+1/x4.​

Answer 1

Solution

Given :-

• x + 1/x = √5 ______(1)

Find :-

• Value of x² + 1/x²
• Value of x⁴ + 1/x⁴

Explanation

Formula

$\dag\boxed{\underline{\tt{\red{\:(a+b)^2\:=\:a^2+b^2+2ab}}}}$

First Calculate x² + 1/x²

For this, Squaring both side of equ(1)

==> (x + 1/x)² ✓ (√5)²

==> x² + (1/x)² + 2x × 1/x = √5 × √5

==> x² + 1/x² + 2 = 5

==> x² + 1/x² = 5 - 3

==.> x² + 1/x² = 2 ________(2)

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Now, Calculate x⁴ + 1/x⁴

For this, squaring both side of equ(2)

==> (x² + 1/x²)² = 2²

==> x⁴ + 1/x⁴ + 2×x² × 1/x² = 4

==> x⁴ + 1/x⁴ + 2 = 4

==> x⁴ + 1/x⁴ = 4 - 2

==> x⁴ + 1/x⁴ = 2

Hence

• Value of x² + 1/x² = 2
• Value of x⁴ + 1/x⁴ = 2

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