If x³ + 7x² + 49x -1 = 0 find x³ + \(\frac{7}{x}\)

344

x³ + 7x² + 49x -1 = 0

here we can see

x³ ⇒ × (\(\frac{7}{x}\)) ⇒ 7x² ⇒ × (\(\frac{7}{x}\)) ⇒ 49x

Therefore, this equation is in G.P. and the common ratio is = \(\frac{7}{x}\) and n = no. of term following G.P. = 3

Now,

First term × \(\frac{ { \left(Common\;ratio \right) }^{n} -1 }{(Common\;ratio - 1)}\) = 1

⇒ x³ × \(\frac{ { \left({ \left(\frac{7}{x}\right) }^{3}-1 \right)}}{\frac{7}{x}-1}\) = 1

⇒ x³ ×\(\frac{ { \left(\frac{343}{x^3}-1 \right)}}{\frac{7}{x}-1}\) = 1

⇒ 343 - x³ = \(\frac{7}{x}\) - 1

⇒ x³ + \(\frac{7}{x}\) = 343 + 1 = 344