Question

if the sum of square of a number and its reciprocal is greater than 7 / 4 the sum of that number and its reciprocal then find that number

Answer 1

Step-by-step explanation:

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Answer 2

Given:

the sum of the square of a number and its reciprocal is greater than 7 / 4 the sum of that number and its reciprocal.

To Find:

The number.

Solution:

Let the number be x,

As the given information,

⇒ x² + 1/x² = x + 1/x + 7/4

{ (x + 1/x)² = x² + 1/x² + 2 }

⇒ (x + 1/x)² = (x + 1/x) + 7/4 + 2

let (x + 1/x) = m

so,

⇒ m² = m + 7/4 + 2

⇒ m² - m - 15/4 = 0

⇒ 4m² -4m - 15 = 0

⇒ 4m² -10m + 6m - 15 = 0

⇒ 2m(2m -5) + 3(2m -5) = 0

⇒ (2m + 3).(2m -5) = 0

⇒ 2m + 3 = 0 and 2m -5 = 0

⇒ m = -3/2  and m = 5/2

as (x + 1/x) = m

then,

(x + 1/x) = 5/2  and (x + 1/x) = -3/2

⇒ 2x² + 2 = -3x

⇒ 2x² + 3x + 2 = 0

roots are not real as b² -4ac < 0.

⇒ 2x² + 2 = 5x          

⇒ 2x² - 5x +2 = 0    ,

⇒  2x² - 4x - x +2 = 0      

⇒ 2x(x - 2) -1(x - 2)=0        

⇒ (2x - 1).(x-2) = 0            

⇒ x = 1/2 , x = 2                

Hence, the number are 2 and 1/2.