The sum of two numbers is #-29#. The product of the same two numbers is #96#. What are the two numbers?

1 Answer
Feb 14, 2018

Answer:

The two numbers are #-4# and #-24#.

Explanation:

You can translate the two statements from English to math:

#stackrel(x+y) overbrace"The sum of two numbers"" " stackrel(=) overbrace"is"" " stackrel(-28) overbrace"-28."#

#stackrel(x*y) overbrace"The product of the same two numbers"" " stackrel(=) overbrace"is"" " stackrel(96) overbrace"96."#

Now we can create a system of equations:

#{(x+y=-28, qquad(1)), (x*y=96, qquad(2)):}#

Now, solve for #x# in equation #(1)#:

#color(white)(=>)x+y=-28#

#=>x=-28-y#

Plug this new #x# value into equation #(2)#:

#color(white)(=>)x*y=96#

#=>(-28-y)*y=96#

#color(white)(=>)-28y-y^2=96#

#color(white)(=>)-y^2-28y-96=0#

#color(white)(=>)y^2+28y+96=0#

#color(white)(=>)(y+24)(y+4)=0#

#color(white)(=>)y=-4,-24#

Lastly, plug both of these #y# values back into equation #(1)#:

For #y=-4#:

#color(white)(=>)x+y=-28#

#=>x-4=-28#

#color(white)(=>)x=-24#

And for #y=-24#:

#=>x-24=-28#

#color(white)(=>)x=-4#

Finally, we see that there are two solutions which are the same: #(-4,-24)# and #(-24,-4)#.

This means that the two numbers are #-4# and #-24#.