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Examples 1 to 3

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(1) If  a + b  =  9  and  a – b  =  5,  find   a2 +  b2   and  ab.

Soln. 2( a2+b2 ) =  (a + b) 2 +  (a – b) 2

=    (9) 2 +  (5) 2

=    106

4ab =  (a + b) 2 –  (a – b) 2

=    (9) 2 –  (5) 2

=    56

\therefore (a2 +b2 )   =    53 \therefore  ab   =    28

(2) If    x – \frac{1}{x}  =  2\frac{2}{3} ,    find

(i)  x + \frac{1}{x}      (ii)  x2 – \frac{1}{x^{2}}     (iii)   x2 + \frac{1}{x^{2}}    (iv)   x3 + \frac{1}{x^{3}}    (v)   x3 – \frac{1}{x^{3}}

Soln. Here,  x – \frac{1}{x}  =  2\frac{2}{3}  = \frac{8}{3}

\left (x+\frac{1}{x} \right )^{2}=   \left (x-\frac{1}{x} \right )^{2}   +  4

=   \left (\frac{8}{3} \right )^{2}   +  4

=   \frac{100}{9}

\therefore x + \frac{1}{x} = \pm \frac{10}{3}x^{2} – \frac{1}{x^{2}} = \left (x+\frac{1}{x} \right ) \times \left (x-\frac{1}{x} \right )

= \left ( \pm \frac{10}{3} \right ) \times \left (\frac{8}{3} \right )

= \pm \frac{80 }{9 }

■   x2 + \frac{1}{x^{2}}  =   \left (x+\frac{1}{x} \right )^{2}   –  2    =    \left (x-\frac{1}{x} \right )^{2}   +   2

■   (a + b) 2  =  (a – b) 2  +  2  [ middle term of  (a + b) 2 =  2ab]

■   (a – b) 2  =  (a + b) 2  +  2  [ middle term of  (a – b) 2  =  – 2ab]

x2 + \frac{1}{x^{2}} = \left (x+\frac{1}{x} \right )^{2} – 2

=   \frac{100}{9}   –  2

=    \frac{82}{9}

x3 + \frac{1}{x^{3}} =  \left (x+\frac{1}{x} \right )^{3}  – 3\left (x+\frac{1}{x} \right )

=  \left (\pm \frac{10}{3} \right )^{3}  – 3\left (\pm \frac{10}{3} \right )

=  \pm \frac{1000}{27} \mp \frac{30}{3}

=    \frac{\pm 1000 \mp 270}{27}

=     \pm \frac{730}{27}

x3 – \frac{1}{x^{3}} = \left (x-\frac{1}{x} \right )^{3} – 3\left (x-\frac{1}{x} \right )

=   \left (\frac{8}{3} \right )^{3} – 3\left (\frac{8}{3} \right )

=    – \frac{152}{27}

(3) If     x3 + \frac{1}{x^{3}}  =   322,   then find the value of  x+\frac{1}{x}.

 (a) 5 (b) 7 (c) 8 (d) 10

Soln. Here,   x3 + \frac{1}{x^{3}}   =  \left (x+\frac{1}{x} \right )^{3}  – 3\left (x+\frac{1}{x} \right )

\therefore  322  = \left (x+\frac{1}{x} \right )^{3} – 3 \left (x+\frac{1}{x} \right )           ––––––––––––––––––––––  (1)

Checking each option in equation  (1), we get

x+\frac{1}{x} =  7

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