How do you solve #10u^3-5u^2=0#?

1 Answer
Jun 2, 2016

Answer:

#u'=0, u''=0, u'''=1/2#

Explanation:

This is a third degree equation without and independent term, therefore it can be solve though factorization. Start by factoring u:

#u(10u^2-5u)=0#.

Notice that, for this new equation to result in 0, any of the factors must me 0. If the first #u# is zero, then we have found one answer: #u'=0#.

However, if #10u^2-5u=0#, then we must factor the equation again to solve it:

#u(10u-5)=0#.

Again, we have a case in which the first #u# can be zero, so:
#u''=0#.

Now, if #10u-5=0#, then:
#10u=5#
#u'''=5/10=1/2#.

Do notice that this method only works when the equation is equals zero and there is no independent term. We could also have taken a shortcut and factored #u^2(10u-5)# right from the beginning, since we have two identical answers and the original equation does not have #u^1#.