What is the solution set for 3x^5-48x=0?

1 Answer
Aug 8, 2015

0, +-2, +-2i

Explanation:

Note that this is polynomial an equation of the 5th degree, so it should have 5 solutions.

3x^5 - 48x = 0
=> 3x (x^4 - 16) = 0
=> x ((x^2)^2 - 4^2) = 0 (Dividing both sides by 3)
=> x (x^2 + 4) (x^2 - 4) = 0 (Since x^2 - y^2 = (x + y)(x - y) )
=> x (x^2 - (-4)) (x^2 - 4) = 0 (*)
=> x (x^2 - (-4)) (x^2 - 4) = 0
=> x (x^2 - (2i)^2) (x^2 - 2^2) = 0 ( i^2 = -1 )
=> x (x + 2i) (x - 2i) (x + 2) (x - 2) = 0
=> x = 0, +-2, +-2i

If you are not looking for complex roots, at the step marked (*), note that x^2 + 4 is always positive for all real values of x, and thus divide by x^2 + 4. Then you can continue in the exact same way as given.