# How do you solve #1 + 8/(x - 5) = 3/x# and find any extraneous solutions?

##### 1 Answer

There are no Real solutions, but there are Complex solutions:

#x = +-sqrt(15)i#

#### Explanation:

Given:

#1+8/(x-5) = 3/x#

Note that neither

Multiply both sides by

#x + (8x)/(x-5) = 3#

Multiply both sides by

#x(x-5) + 8x = 3(x-5)#

Expand both sides to get:

#x^2-5x+8x = 3x-15#

which simplifies to:

#x^2+3x = 3x-15#

Subtract

#x^2 = -15#

This has no Real solutions since

If we are interested in Complex solutions, then add

#0 = x^2+15 = x^2 - (sqrt(15)i)^2 = (x-sqrt(15)i)(x+sqrt(15)i)#

where *imaginary unit*, with the property that

Hence solutions