Question

$\\ if \: \frac{a}{b} + \frac{b}{a} = - 1 \: then \: {a}^{3} - {b}^{3} = {?}$


​

Answer 1

$\huge \green{given : } \: \\ \: \: \: \: \: \: \frac{a}{b} + \frac{b}{a} = - 1$

$\huge \green{to \: find : } \\ \: \: \: \: \: \: \: \: \: {a}^{3} - {b}^{3} = {?}$

$\huge \green{solution : }$

$\sf \: \: \: \: \implies \: \frac{a}{b} + \frac{ b}{a} = - 1$

$\sf \: \: \: \: \: \implies \frac{ { a}^{2} + {b}^{2} }{ba} = - 1$

$\sf \: \: \: \: \: \: \boxed{ \implies {a}^{2} + {b}^{2} = - ab }$

$\huge \red{Equation : }$

$\bf \: ( {a}^{3} - {b}^{3} ) = \blue{ (a - b)( {a}^{2} + ab + {b}^{2} )}$

$\bf \: \: \: \: \: \: \: \: \: \: \: \: \: \blue{ = (a - b)( {a}^{2} + {b}^{2} + ab)}$

$\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \blue{(a - b)( - \cancel{ab} + \cancel{ab})}$

$\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \blue{ = (a - b) - 1}$

$\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \blue{ = 0}$

$\green{Hence \: proved }$

$\: \: \sf \boxed{ {a}^{3} - {b}^{3} = - 1 }$

Answer 2

Answer:

given:

b

a

+

a

b

=−1

\begin{gathered} \huge \green{to \: find : } \\ \: \: \: \: \: \: \: \: \: {a}^{3} - {b}^{3} = {?}\end{gathered}

tofind:

a

3

−b

3

=?

\huge \green{solution : }solution:

\begin{gathered} \sf \: \: \: \: \implies \: \frac{a}{b} + \frac{ b}{a} = - 1 \\ \end{gathered}

⟹

b

a

+

a

b

=−1

\begin{gathered} \sf \: \: \: \: \: \implies \frac{ { a}^{2} + {b}^{2} }{ba} = - 1 \\ \end{gathered}

⟹

ba

a

2

+b

2

=−1

\sf \: \: \: \: \: \: \boxed{ \implies {a}^{2} + {b}^{2} = - ab }

⟹a

2

+b

2

=−ab

\huge \red{Equation : }Equation:

\bf \: ( {a}^{3} - {b}^{3} ) = \blue{ (a - b)( {a}^{2} + ab + {b}^{2} )}(a

3

−b

3

)=(a−b)(a

2

+ab+b

2

)

\bf \: \: \: \: \: \: \: \: \: \: \: \: \: \blue{ = (a - b)( {a}^{2} + {b}^{2} + ab)}=(a−b)(a

2

+b

2

+ab)

\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \blue{(a - b)( - \cancel{ab} + \cancel{ab})}=(a−b)(−

ab

+

ab

)

\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \blue{ = (a - b) - 1}=(a−b)−1

\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \blue{ = 0}=0

\green{Hence \: proved }Henceproved

\: \: \sf \boxed{ {a}^{3} - {b}^{3} = - 1 }

a

3

−b

3

=−1

hope it helps you ✌️