Question

if a+¹/a=1 then a³+1=?​

Answer 1

$\underline{ \underline{ \large{ \red{ \bf{Given}}}}}$

$\mathrm{ a + \frac{1}{a} = 1 }$

$\underline{ \underline{ \large{ \red{ \bf{To \: find}}}}}$

$\mathrm{{a}^{3} + 1 = ?}$

$\underline{ \underline{ \large{ \red{ \bf{Solution}}}}}$

we have

$\mathrm{a + \frac{1}{a} = 1 }$

taking LCM

$\mathrm{ \frac{ {a}^{2} + 1 }{a} = 1}$

cross multiplying

$\mathrm{ {a}^{2} + 1 = a}$

$\mathrm{ {a}^{2} + 1 - a = 0 \: ....eqn(1)}$

Now,

we have to find

$\mathrm{ {a}^{3} + 1}$

$\mathrm{( {a})^{3} + ( {1})^{3} }$

using identity

$\small{\bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} + {y}^{2} - xy) }}$

we will get

$\mathrm{(a + 1)( {a}^{2} + 1 - a) }$

using eqn (1) we will get

$\mathrm{(a + 1)(0) = 0}$

Hence,

$\boxed{ \pink {\bf{ {a}^{3} + 1 = 0}}}$

Answer 2

Given  : a  + 1/a)   = 1

To Find  : a³ + 1

Solution:

a  + 1/a   = 1

=> a²  + 1  = a

=> a² - a  +  1 = 0

a³ + b³  = (a + b)(a² -ab + b²)

a = a  and b = 1

a³ + 1  = (a + 1)(a² - a + 1)

substituting a² - a  +  1 = 0

=> a³ + 1  = (a + 1)0

=> a³ + 1  =  0

Hence if   a  + 1/a   =  1  then  a³ + 1 = 0

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