Question

If z+1/z=5,find the value of z-1/z​

Answer 1

Solution :-

Provided  

$\circ \sf { z + \dfrac{1}{z} = 5}$

We have to find the value of

$\circ \sf { z - \dfrac{1}{z} }$

Now as we know

$\circ \sf { (a +b)^2 - 4ab = (a-b)^2}$

Therefore by using this

$\implies \sf {( z + \dfrac{1}{z})^2 - 4 = \left( z - \dfrac{1}{z}\right)^2}$

$\implies \sf{(5)^2 - 4 = \left( z - \dfrac{1}{z}\right)^2}$

$\implies \sf {25 - 4 = \left( z - \dfrac{1}{z}\right)^2}$

$\implies \sf {21 = \left( z - \dfrac{1}{z}\right)^2}$

By taking square root

$\implies \sf { z - \dfrac{1}{z} = \sqrt{21}}$

Answer 2

$\sf{\underline{\boxed{\mathfrak{Given}}}}$

• $\sf z + \dfrac{1}{z} = 5$

$\sf{\underline{\boxed{\mathfrak{To\: Find}}}}$

• $\sf z - \dfrac{1}{z} = ????$

$\sf{\underline{\boxed{\mathfrak{solution}}}}$

✰$\sf{\underline{\boxed{\normalsize{( x + \dfrac{1}{x} )^2 - (x - \dfrac{1}{x})^2= 4 }}}}$

➤$5^2 - ( z + \dfrac{1}{z})^2 = 4$

➤$25 - ( z + \dfrac{1}{z})^2 = 4$

➤$( z + \dfrac{1}{z})^2 = 25 - 4$

➤$( z + \dfrac{1}{z})^2 = 21$

➤$z + \dfrac{1}{z} = \sqrt{ 21}$

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$\sf{\underline{\boxed{\normalsize{z + \dfrac{1}{z} = \sqrt {21}}}}}$

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