Question

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

Answer 1

`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`.