Question

 If x + (1/x) = 5, find the value of [x - (1/x)]^2.​

Answer 1

Step-by-step explanation:

Given:

$\rightarrowtail \bold{x+\dfrac{1}{x}=5 }$

To find:

$\rightarrowtail \bold{(x-\dfrac{1}{x})^{2} }$

Solution:

• Squaring on both sides.

$\mapsto \bold{(x+\dfrac{1}{x})^{2}=5^{2} }$

• ∵ (a + b)² = a² + b² + 2ab.

$\mapsto \bold{x^{2}+\dfrac{1}{x^{2}}+2(x)(\dfrac{1}{x})=25 }$

$\mapsto \bold{x^{2}+\dfrac{1}{x^{2}}+2=25 }$

$\Rrightarrow \bold{x^{2}+\dfrac{1}{x^{2}}=23 }$

• ∵ (a - b)² = a² + b² - 2ab

$\mapsto \bold{x^{2}+\dfrac{1}{x^{2}}-2(x)(\dfrac{1}{x}) }$

• ∵ x² + 1/x² = 23

$\twoheadrightarrow \bold{23-2}$

$\twoheadrightarrow \bold{25}$

• ∴ 25 is the value of x - 1/x²