Question

If x^2 -5x + 1 = 0, find x^10 + 1/x^5 has the value ?
A) 2524 (B) 2525
(C) 2424 (D) 2010

Answer 1

Answer:

(B) 2525 is the correct answer

Step-by-step explanation:

Given :

x² - 5x + 1 = 0

To find :

The value of  $\mathrm{\dfrac{x^{10} + 1}{x^5}}$ from the options

(A) 2524

(B) 2525

(C) 2424

(D) 2010

Solution :

x² - 5x + 1 = 0

=> x² + 1 = 5x

Dividing both sides by x

$\implies \mathrm{\dfrac{x^2+1}{x} = 5 }$

$\implies \mathrm{x + \dfrac{1}{x} = 5 }$   ____ [1]

Squaring on both sides, we get

$\implies \mathrm{\bigg(x + \dfrac{1}{x} \bigg)^2= 5^2 }$

$\implies \mathrm{x^2 + \dfrac{1}{x^2} + 2 = 25 }$

$\implies \mathrm{x^2 + \dfrac{1}{x^2} = 25 -2 }$

$\implies \mathrm{x^2 + \dfrac{1}{x^2} = 23 }$  ____ [2]

Cubing [1] on both sides

$\implies \mathrm{\bigg(x + \dfrac{1}{x}\bigg)^3 = 5^3 }$

$\implies \mathrm{x^3 + 3x + \dfrac{3}{x} + \dfrac{1}{x^3} = 125 }$

$\implies \mathrm{x^3 + 3\bigg(x + \dfrac{1}{x} \bigg)+ \dfrac{1}{x^3} = 125 }$

$\implies \mathrm{x^3 + 3(5)+ \dfrac{1}{x^3} = 125 }$  ____ [From [1]]

$\implies \mathrm{x^3 + \dfrac{1}{x^3} = 125 -15 }$

$\implies \mathrm{x^3 + \dfrac{1}{x^3} = 110 }$  ____ [3]

Now, multiply [2] and [3]

$\implies \mathrm{\bigg(x^2 + \dfrac{1}{x^2}\bigg)\bigg(x^3 + \dfrac{1}{x^3} \bigg)= 23 \times 110}$

$\implies \mathrm{\bigg(x^5 + x + \dfrac{1}{x} + \dfrac{1}{x^5} \bigg)= 2530}$

From [1],

$\implies \mathrm{\bigg(x^5 + 5 + \dfrac{1}{x^5} \bigg)= 2530}$

$\implies \mathrm{x^5 + \dfrac{1}{x^5}= 2530 - 5}$

$\implies \mathrm{x^5 + \dfrac{1}{x^5}= 2525}$

$\therefore \underline{\boxed{\mathbf{\dfrac{x^{10} + 1}{x^5}= 2525}}}$

Hope it helps!!

Answer 2

Answer:

If x^2 -5x + 1 = 0, find x^10 + 1/x^5 has the value ?

A) 2524 (B) 2525

(C) 2424 (D) 2010

Step-by-step explanation: