Question

The equation 3x^4 - 9x^3 - 3x^2 = 0 has three real solutions.

they are... ? ___________, _________, _________.

hint: factor out common factor(yielding one solution), then maybe use quadratic formula to find the others.

Answer 1

0 is one solution.

Answer 2

that's what i put for one..

Answer 3

couldn't find the other 2 :/

Answer 4

Factor out the x = 0 solution:

3x^4 - 9x^3 - 3x^2 = 3x^2 ( x^2 - 3x - 1 )

Now that equation is equal to zero. So ...

Answer 5

What would i have to do after that? because thats what i got for factoring it out the first time..

Answer 6

Yes, and now this equation is equal to zero if and only if x = 0
or
x^2 - 3x - 1 = 0

You know how to solve this last equation, and the roots of THAT equation are also roots of your original equation.

That's why we bother with this procedure.

So, what are the roots of x^2 - 3x - 1 = 0?

Answer 7

what i got for the roots were..

(sqrt(13)+3)/2
-(sqrt(13)+3)/2

Answer 8

\[(\sqrt(13)+3)/2\]

Answer 9

careful with signs; almost right.

Answer 10

\[-(\sqrt(13)+3)/2\]

Answer 11

This last you've written down is not a root.

Answer 12

so what did i do wrong?

Answer 13

ive written down for the answers.
0, (sqrt(13)+3)/2, & -(sqrt(13)+3)/2

Answer 14

but they said it was incorrect

Answer 15

\[\frac{3 \pm \sqrt{13}}{2}\]

Answer 16

yeah, be careful with -(sqrt(13)+3)/2

Answer 17

it should be -(sqrt(13)-3)/2