Question

Find the value of x²+ 1/x², if x-1/x =√3
​

Answer 1

Question :

Find the value of x² + 1/x², if x-1/x =√3

​

Answer :

$\boxed{\bf x^2+\frac{1}{x^2}=5}$

Given :

$x-\frac{1}{x} =\sqrt{3}$

To find :

$x^{2} + \frac{1}{x^2} = \ ?$

Solution :

 ${x-\frac{1}{x}=\sqrt{3}}$

Squaring on both sides,

   

$(x-\frac{1}{x})^2=\sqrt{3}^2 \\\\ x^{2} +(\frac{1}{x})^2-2(x)(\frac{1}{x})=3 \\\\ x^{2} +\frac{1}{x^2}-2=3 \\\\ x^{2} +\frac{1}{x^2}=3+2\\\\x^{2} +\frac{1}{x^2}=5$

-------------------------------------

Identity used :

$(a-b)^2=a^2+b^2-2ab$

-------------------------------------

OTHER IDENTITIES :

$(a+b)^2=a^2+b^2+2ab \\\\ a^2-b^2=(a+b)(a-b)\\\\ (a+b)^3=a^3+b^3+3a^2b+3ab^2\\\\ (a-b)^3=a^3-b^3-3a^2b+3ab^2$

Answer 2

Question :

Find the value of x² + 1/x², if x-1/x =√3

Answer :

\boxed{\bf x^2+\frac{1}{x^2}=5}

x

2

+

x

2

1

=5

Given :

x-\frac{1}{x} =\sqrt{3}x−

x

1

=

3

To find :

x^{2} + \frac{1}{x^2} = \ ?x

2

+

x

2

1

= ?

Solution :

{x-\frac{1}{x}=\sqrt{3}}x−

x

1

=

3

Squaring on both sides,

\begin{gathered}(x-\frac{1}{x})^2=\sqrt{3}^2 \\\\ x^{2} +(\frac{1}{x})^2-2(x)(\frac{1}{x})=3 \\\\ x^{2} +\frac{1}{x^2}-2=3 \\\\ x^{2} +\frac{1}{x^2}=3+2\\\\x^{2} +\frac{1}{x^2}=5\end{gathered}

(x−

x

1

)

2

=

3

2

x

2

+(

x

1

)

2

−2(x)(

x

1

)=3

x

2

+

x

2

1

−2=3

x

2

+

x

2

1

=3+2

x

2

+

x

2

1

=5

-------------------------------------

Identity used :

(a-b)^2=a^2+b^2-2ab(a−b)

2

=a

2

+b

2

−2ab

-------------------------------------

OTHER IDENTITIES :

\begin{gathered}(a+b)^2=a^2+b^2+2ab \\\\ a^2-b^2=(a+b)(a-b)\\\\ (a+b)^3=a^3+b^3+3a^2b+3ab^2\\\\ (a-b)^3=a^3-b^3-3a^2b+3ab^2\end{gathered}

(a+b)

2

=a

2

+b

2

+2ab

a

2

−b

2

=(a+b)(a−b)

(a+b)

3

=a

3

+b

3

+3a

2

b+3ab

2

(a−b)

3

=a

3

−b

3

−3a

2

b+3ab

2