Question

Without determining the value of x, evaluate x² + ¹/x² if x - ¹/x = 5​

Answer 1

Given: {x -(1/x)} = 5

Asked: = {x² + (1/x²)} = ?

Solution:

Given that:

{x - (1/x)} = 5

On squaring both sides then

→ {x - (1/x)}² = (5)²

Now

Compare the LHS with (a-b)², we get

a = x and b = 1/x

Using identity (a -b)² = a² - 2ab + b² , we get

→ {x - (1/x)}² = (5)²

→ x² - 2(x)(1/x) + (1/x)² = 5*5

Multiply the number on RHS.

→ x² - 2(x)(1/x) + (1/x²) = 25

Cancel both “x” on RHS. Because they are in multiple sign.

→ x² - 2 + (1/x²) = 25

Shift the number 2 from LHS to RHS, changing it's sign.

→ x² + (1/x²) = 25 + 2

Add the numbers on RHS.

→ x² + (1/x²) = 27

Therefore, {x² + (1/x²)} = 27

Answer: Hence, the value of {x² + (1/x²)} = 27

Please let me know if you have any other questions.

Answer 2

Given,

`x - (1)/(x) = 5`

Squaring both sides, we get

`= > (x - (1)/(x) )^(2) = {(5)}^(2)`

By using the identity: (a - b)² = a² - 2ab + b², we have:

`= > {x}^(2) - 2(x)( (1)/(x) ) + ((1)/(x) ) ^(2) = 25 \\ = > {x}^(2) - 2 + \frac{1}{ {x}^(2) } = 25`

Now, transpose -2 to the right hand side.

`= > {x}^(2) + \frac{1}{ {x}^(2) } = 25 + 2 \\ = > {x}^(2) + \frac{1}{ {x}^(2) } = 27`

Answer:

27

Hope you could understand.

If you have any query, feel free to ask.