Question

Given - x^2-3x+1=0 . Find x^5+1/x^5 ​

Answer 1

Answer:

x² - 3x + 1 = 0

Simplifying the given equation,

➸ (x² - 3x + 1 = 0)/x

➸ x - 3 + 1/x = 0/x

➸ x - 3 + 1/x = 0

➸ x + 1/x = 3

Here,

➸ $\tt{{\bigg(x+\dfrac{1}{x}\bigg)}^2=3^3}$

➸ $\tt{\bigg(x^2+\dfrac{1}{x^2}+2\times x^2\times\dfrac{1}{x^2}\bigg)=9}$

➸ $\tt{\bigg(x^2+\dfrac{1}{x^2}+2\bigg) =9}$

➸ $\tt{\bigg(x^2+\dfrac{1}{x^2}\bigg)=7}$

Also, similarly, x⁴+1/x⁴=47.

Now,

Multiplying x+1/x on both sides.

➸ (x⁴+1/x⁴)(x+1/x)=47(x+1/x)

Simplifying,

➸ x^5 + x³ + 1/x³ + 1/x^5 = 47*3

➸ x^5 + 1/x^5 = 141 - (x³ + 1/x³)

➸ x^5 + 1/x^5 = 141 - [(x+ 1/x)(x²+ 1/x²-1)]

[By identity a³+b³ = (a+b)(a²+b²-ab)]

➸ x^5 + 1/x^5 = 141 - [(3)(7-1)]

➸ x^5 + 1/x^5 = 141 - 18

➸ x^5 + 1/x^5 = 123

Answer 2

Answer:

x² - 3x + 1 = 0

Simplifying the given equation,

➸ (x² - 3x + 1 = 0)/x

➸ x - 3 + 1/x = 0/x

➸ x - 3 + 1/x = 0

➸ x + 1/x = 3

similarly, x⁴+1/x⁴=47.

Now,

Multiplying x+1/x on both sides.

➸ (x⁴+1/x⁴)(x+1/x)=47(x+1/x)

Simplifying,

➸ x^5 + x³ + 1/x³ + 1/x^5 = 47*3

➸ x^5 + 1/x^5 = 141 - (x³ + 1/x³)

➸ x^5 + 1/x^5 = 141 - [(x+ 1/x)(x²+ 1/x²-1)]

[By identity a³+b³ = (a+b)(a²+b²-ab)]

➸ x^5 + 1/x^5 = 141 - [(3)(7-1)]

➸ x^5 + 1/x^5 = 141 - 18

➸ x^5 + 1/x^5 = 123