Question

Plz solve this question in copy fast!

Answer 1

Given x = $\frac{5 - \sqrt{21} }{2}$

⇒ 1/x = $\frac{2}{5 - \sqrt{21} }$

multiply the numerator and denominator of 1/x with (5 + √21) then we get :

⇒ 1/x = $\frac{2(5 + \sqrt{21}) }{(5 - \sqrt{21})(5 + \sqrt{21} ) }$

we know that (a + b)(a - b) = a² - b² ⇒ the above expression becomes :

⇒ 1/x = $\frac{2(5 + \sqrt{21}) }{25 - 21}$

⇒ 1/x = $\frac{5 + \sqrt{21} }{2}$

⇒ x + 1/x = $\frac{5 - \sqrt{21} }{2} + \frac{5 + \sqrt{21} }{2}$

⇒ x + 1/x = 10/2 = 5

we know that (x + 1/x)³ = x³ + 1/x³ + 3x².1/x + 3x.1/x²

⇒ (x + 1/x)³ = x³ + 1/x³ + 3x + 3/x

⇒ (x + 1/x)³ = x³ + 1/x³ + 3(x + 1/x)

⇒ x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)

we know that the value of (x + 1/x) = 5

⇒ x³ + 1/x³ = 5³ - 3×5 = 125 - 15 = 110

we know that (x + 1/x)² = x² + 1/x² + 2

⇒ x² + 1/x² = (x + 1/x)² - 2

⇒  x² + 1/x² = 5² - 2 = 23

substituting all the values we got in the question we get :

⇒ 110 - 5×23 + 5

⇒ 115 - 115 = 0