Question

9. If x^2 + 4x -1 = 0 and x is positive then find the value of
(a) x + 1 (b) x² + 1/x^2.​

Answer 1

Correct Question:

If x² + 4x - 1 = 0 , and x is positive then find the value of ;

a). x + 1/x

b). x² + 1/x²

Answer:

a). x + 1/x = 2√5

b). x² + 1/x² = 18

Solution:

We have ;

=> x² + 4x - 1 = 0

=> x² + 4x + 2² - 2² - 1 = 0

=> x² + 2•x•2 + 2² = 2² + 1

=> (x + 2)² = 5

=> x + 2 = √5

=> x = √5 - 2 ( x > 0 )

Now,

=> x = √5 - 2

=> 1/x = 1/(√5 - 2)

=> 1/x = (√5 + 2)/(√5 - 2)(√5 + 2)

=> 1/x = (√5 + 2)/[ (√5)² - 2² ]

=> 1/x = (√5 + 2)/(5 - 4)

=> 1/x = (√5 + 2)/1

=> 1/x = √5 + 2

Now,

=> x + 1/x = √5 - 2 + √5 + 2

=> x + 1/x = 2√5

Hence,

x + 1/x = 2√5

Now,

=> x + 1/x = 2√5

=> (x + 1/x)² = (2√5)²

=> x² + 2•x•(1/x) + (1/x)² = 20

=> x² + 2 + 1/x² = 20

=> x² + 1/x² = 20 - 2

=> x² + 1/x² = 18

Hence,

x² + 1/x² = 18