Question

if x =(2+root 5) ,find the value of x+1/x

Answer 1

Given:

The value of x = ( 2+ $\sqrt{5}$ )

To find:

The value of x+(1/x)

Solution:

The value of x is equal to (2+ $\sqrt{5}$ ).

Step 1:

We will directly put the value of x in x+(1/x).

x + (1/x) = $(2+\sqrt{5}) + ( \frac{1}{2+\sqrt{5} })$

The denominators are different,

therefore, the adding of the values is not possible without taking the L.C.M

Step 2:

We will take L.C.M. and make the denominators the same.

L.C.M of 1 and ( 2+$\sqrt{5}$ )  is (2+$\sqrt{5}$).

Step 3:

Since the denominators are the same, we can add the values.

x + (1/x) = { ( 2+ $\sqrt{5}$ ) ( 2+ $\sqrt{5}$  )  + 1 } ÷ (2+ $\sqrt{5}$ )

In the numerator, ( 2 + √5 ) ( 2 + √5 ) is comparable to the following algebraic identity,

( a+b) (a+b) = ( a+ b)² = a² + b² + 2ab

( 2 + √5 ) ( 2 + √5 ) = ( 2 + √5 ) ²

= 2² + (√5)² + 2( 2) ( √5)

= 4 + 5 + 4√5

= 9 + 4√5

x + (1/x) =  (9 + 4√5 + 1 ) / ( 2 + √5)

x + (1/x) = ( 10 + 4√5 ) / ( 2 + √5)

Step 4:

To simplify the above terms, we will rationalize the fraction.

Multiplying numerator and denominator by ( 2 - √5)

= ( 10 + 4√5 )( 2 - √5) / ( 2 + √5) ( 2 - √5)

The denominator ( 2 + √5) ( 2 - √5) is in the form of identity ( a+b) (a-b)

Since , ( a+b) (a-b) = ( a² - b²)

∴ ( 2 + √5) ( 2 - √5) = (4 - 5)

x + (1/x) = (20 - 10 √5 + 8√5 - 20) / ( 4 - 5)

On solving the above equations, we get,

x + (1/x) = 2√5

The value of x + (1/x) is 2√5.

Answer 2

Answer:

Given:

The value of x = ( 2+ \sqrt{5}

5

)

To find:

The value of x+(1/x)

Solution:

The value of x is equal to (2+ \sqrt{5}

5

).

Step 1:

We will directly put the value of x in x+(1/x).

x + (1/x) = (2+\sqrt{5}) + ( \frac{1}{2+\sqrt{5} })(2+

5

)+(

2+

5

1

)

The denominators are different,

therefore, the adding of the values is not possible without taking the L.C.M

Step 2:

We will take L.C.M. and make the denominators the same.

L.C.M of 1 and ( 2+\sqrt{5}

5

) is (2+\sqrt{5}

5

).

Step 3:

Since the denominators are the same, we can add the values.

x + (1/x) = { ( 2+ \sqrt{5}

5

) ( 2+ \sqrt{5}

5

) + 1 } ÷ (2+ \sqrt{5}

5

)

In the numerator, ( 2 + √5 ) ( 2 + √5 ) is comparable to the following algebraic identity,

( a+b) (a+b) = ( a+ b)² = a² + b² + 2ab

( 2 + √5 ) ( 2 + √5 ) = ( 2 + √5 ) ²

= 2² + (√5)² + 2( 2) ( √5)

= 4 + 5 + 4√5

= 9 + 4√5

x + (1/x) = (9 + 4√5 + 1 ) / ( 2 + √5)

x + (1/x) = ( 10 + 4√5 ) / ( 2 + √5)

Step 4:

To simplify the above terms, we will rationalize the fraction.

Multiplying numerator and denominator by ( 2 - √5)

= ( 10 + 4√5 )( 2 - √5) / ( 2 + √5) ( 2 - √5)

The denominator ( 2 + √5) ( 2 - √5) is in the form of identity ( a+b) (a-b)

Since , ( a+b) (a-b) = ( a² - b²)

∴ ( 2 + √5) ( 2 - √5) = (4 - 5)

x + (1/x) = (20 - 10 √5 + 8√5 - 20) / ( 4 - 5)

On solving the above equations, we get,

x + (1/x) = 2√5

The value of x + (1/x) is 2√5.