Question

4. If x2 +1/x2= 27 , find the values of each of the following:
(i)x+1/x
(ii)x-1/x​

Answer 1

★ɢɪᴠᴇɴ:-

• $\sf x^2 + \dfrac{1}{x^2}=27$

★ᴛᴏ ғɪɴᴅ:-

• $\sf x + \dfrac{1}{x}$

• $\sf x-\dfrac{1}{x}$

★ՏϴᏞႮͲᏆϴΝ:-

★ $\sf x + \dfrac{1}{x}$

We know that,

$\qquad\qquad\qquad\bigstar\underline{\boxed{ \sf (a+b)^2 = a^2 + b^2 +2ab}}\bigstar$

Using this identity,

$\implies\sf \Bigg \lgroup x +\dfrac{1}{x} \Bigg \rgroup^2 = x^2 + \dfrac{1}{x^2}+2 \times x \times \dfrac{1}{x}$

$\implies\sf \Bigg \lgroup x +\dfrac{1}{x} \Bigg \rgroup^2 = 27+2$

$\implies\sf \Bigg \lgroup x +\dfrac{1}{x} \Bigg \rgroup^2 = 29$

$\implies\sf x +\dfrac{1}{x} = \sqrt{29}$

$\qquad\qquad\qquad\qquad\bigstar \underline{\boxed{\sf x +\dfrac{1}{x} = \sqrt{29}}} \bigstar$

________________________________

★ $\sf x - \dfrac{1}{x}$

We know that,

$\qquad\qquad\qquad\bigstar\underline{\boxed{ \sf (a-b)^2 = a^2 + b^2 -2ab}}$★

Using this identity,

$\implies\sf \Bigg \lgroup x -\dfrac{1}{x} \Bigg \rgroup^2 = x^2 + \dfrac{1}{x^2}-2 \times x \times \dfrac{1}{x}$

$\implies\sf \Bigg \lgroup x -\dfrac{1}{x} \Bigg \rgroup^2 = 27-2$

$\implies\sf \Bigg \lgroup x -\dfrac{1}{x} \Bigg \rgroup^2 = 25$

$\implies\sf x -\dfrac{1}{x} = \sqrt{25}$

$\implies\sf x -\dfrac{1}{x}= 5$

$\qquad\qquad\qquad\qquad\bigstar\underline{\boxed{ \sf x - \dfrac{1}{x} = 5}}\bigstar$

Hope it helps