Question

If x = 9 + 4√5, find the value of 1. X +1/X steps also​

Answer 1

Answer:

9+4√5+ 1/9+4√5

9+4√5 + 1/9+4√5×9-4√5/9-4√5

9+4√5+9-4√5/81-16×5

9+4√5+9-4√5/1

9+9

18

Answer 2

Step-by-step explanation:

→ x = 9 - 4√5

→ 1/x = 1/(9 - 4√5)

Notice that the solving 1/(9 - 4√5) is tougher than just adding two numbers. To simply this, we rationalize the denominator.

Rationalization: multiplying as well as dividing the number by the conjugate of that number.

conjugate: let there be a number a + √b, then its conjugate is a - √b.

Here we are rationalizing 9 - 4√5, for which we multi. and div. it by 9 + 4√5(conjugate of 9 - 4√5).

Rationalize the right hand side :

=> 1/x = 1/(9 - 4√5) × (9 + 4√5)/(9 + 4√5)

=> 1/x = (9 + 4√5)/(9 + 4√5)(9 - 4√5)

Using (a + b)(a - b) = a² - b², for denominator.

=> 1/x = (9 + 4√5)/(9² - (4√5)²)

=> 1/x = (9 + 4√5)/(81 - 80)

=> 1/x = 9 + 4√5

Therefore,

x + 1/x = (9 - 4√5) + (9 + 4√5)

x + 1/x = 9 + 9 = 18

Rationalization is used for convenience. This can be solved without that as well.

= x + 1/x

= (9 - 4√5) + 1/(9 - 4√5)

= { (9 - 4√5)² + 1 }/(9 - 4√5)

= (81 + 80 - 72√5 + 1)/(9 - 4√5)

= (162 - 72√5)/(9 - 4√5)

= 18(9 - 4√5)/(9 - 4√5)

= 18