Question

find x-1/x and x²-1/x² if x+1/x=√5​

Answer 1

Answer:

Here is your answer

Step-by-step explanation:

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Answer 2

Answer:

x - (1/x) = 1 or (-1)

x² - (1/x²) = √5 or (-√5)

Step-by-step explanation:

We are given,

x + (1/x) = √5 ----- 1

Now, Squaring both sides,

(x + (1/x))² = (√5)²

(a + b)² = a² + 2ab + b²

So,

x² + (2 × x × (1/x)) + (1/x)² = 5

x² + 2 + (1/x²) = 5

Thus,

x² + (1/x²) = 5 - 2

x² + (1/x²) = 3

Now,

we know that,

(a - b)² = a² - 2ab + b²

So,

(x - (1/x))² = x² - (2 × x × (1/x)) + (1/x²)

(x - (1/x))² = x² - 2 + (1/x²)

We have,

x² + (1/x²) = 3

Now,

Subtracting 2 from both sides,

x² + (1/x²) - 2 = 3 - 2

x² - 2 + (1/x²) = 1

Hence, from above,

(x - (1/x))² = 1

x - (1/x) = ± √1

x - (1/x) = ± 1

Thus,

x - (1/x) = 1 or (-1) ----- 2

Now, we know that,

a² - b² = (a - b)(a + b)

So,

x² - (1/x²) = (x - (1/x))(x + (1/x))

From eq.1 and eq.2,

Case 1

x² - (1/x²) = (1)(√5)

x² - (1/x²) = √5

Case 2

x² - (1/x²) = (-1)(√5)

x² - (1/x²) = (-√5)

Hence,

x - (1/x) = 1 or (-1)

x² - (1/x²) = √5 or (-√5)

Hope it helped and believing you understood it........All the best