Two fractions are such that their product is -4 and sum is $$\frac{{ - 32}}{{15}}.$$  Find the two fractions.

Solution (By Examveda Team)

$$\eqalign{ & {\text{Let the fractions }}a{\text{ and }}b \cr & ab = - 4 \cr & a + b = \frac{{ - 32}}{{15}}{\text{ }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}\left( 1 \right) \cr & {\left( {a + b} \right)^2} = {\left( {a - b} \right)^2} + 4ab \cr & {\left( {\frac{{ - 32}}{{15}}} \right)^2} = {\left( {a - b} \right)^2} + 4\left( { - 4} \right) \cr & {\left( {a - b} \right)^2} = \frac{{1024}}{{225}} + 16 \cr & a - b = \sqrt {\frac{{4624}}{{225}}} \cr & a - b = \frac{{68}}{{15}}{\text{ }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}\left( 2 \right) \cr & {\text{Equate equation }}\left( 1 \right){\text{ and }}\left( 2 \right){\text{,}}\,{\text{we get}} \cr & a = \frac{{18}}{{15}} = \frac{6}{5} \cr & b = \frac{{ - 100}}{{30}} = \frac{{ - 10}}{3} \cr} $$