Question

Solve the equation 6x^4-11x^3-26x^2+22x+24 = 0 given that the product of two of the roots is equal to the product of the other two.

Answer 1

well the product of all 4 roots is 4 yes?

Answer 2

I'm assuming you're thinking alpha x beta x gamma x delta = 4...

Answer 3

product of 2 of the roots =2 or -2

Answer 4

I need all FOUR roots, since it's a quartic.

Answer 5

yeah but you still have lots of choices. especially since no one says they are rational.

Answer 6

they're not. But there's only 4 possible roots to this equation, given that condition.

Answer 7

(well, at least I know 2 of them aren't.)

Answer 8

This might be a good place to start: http://home.windstream.net/okrebs/page26.html

Particularly rule 5. :)

Answer 9

you could try listing the possible rational roots and then seeing which pair multiplies to 2 or -2. btw the roots are
\[\frac{4}{3},-\frac{3}{2}, 1+\sqrt{3},1-\sqrt{3}\] but i got that by cheating

Answer 10

lol I wanted to see whether this site could give some help, yup, those are the roots though. Alrighty then, I know how to do it, just wondering whether anyone knew how to do it by hand though :P

Answer 11

yeah not really sure how that helps. you are supposed to use the fact that if the roots are
\[r_1,r_2,r_3,r_4\] then
\[r_1r_2=r_3r_4\]

Answer 12

mathteacher's on the right track...lol.

Answer 13

and also that
\[r_1r_2r_3r_4=4\]

Answer 14

there are 4 y-intercepts:

x=-3/2
x= 4/3
x=1-sqrt3
x=1+sqrt3

Answer 15

maybe elementary symmetric functions?

Answer 16

It can also be expressed as:

\[(2 x+3) (3 x-4) (x^2-2 x-2) = 0\]

Answer 17

llort how did u solve it?
did u actually graph it??

Answer 18

\[11=r_1r_2+r_1r_3+r_1r_4+r_2r_3+r_2r_4+r_3r_4\]

Answer 19

no that doesn't get it because i don't know if the the products are 2 or -2.

Answer 20

lol, okay. Was just wondering whether you guys could help out with my maths or not. At least I can rely on you guys to verify my answers ^_^ I was looking for working btw, which is using the sums/products of roots such that you got 4 equations and then solving them all simultaneously, while alsong adhering to the condition.

i.e.
Let the equation is expressed in the form of \[ax^4 + bx^3 + cx^2 + dx + e\]

\[-b/a = r1 + r2 + r3 + r4\]
\[c/a = r1r2 + r1r3+ r1r4 + r2r3 +r2r4 +r3r4\]
\[-d/a = r1r2r3 + r1r3r4 + r1r2r4 + r2r3r4\]
\[e/a = r1r2r3r4\]
\[r1r2 = r3r4\]

therefore...

Then substituting in the values, and working them all out. :)

Answer 21

yeah I graphed it, I used wolfram.

The sum of the roots is 11/6

Answer 22

yeah it is 11/6 not 11 as i wrote. have to use all elementary symmetric functions i thought there was a snap trick.

Answer 23

Just wondering, what topic/class does this fall into in your education systems? For me it's senior mathematics extension.