Question

Please, please, please explain where I went wrong.

Answer 1

Let a, b, c be real numbers such that \(a + b + c=6\) and \(ab + bc + ca = 9.\) If exactly one root of the equation \(x^2 - ( m+2)x + 5m = 0 \) lies between the minimum and maximum value of \(c\), find the number of possible integral values of \(m\).

So \(a^2 + b^2 + c^2 = 18\) according to the question, meaning that the minimum value of \(c\) should be \(-\sqrt{18}\) and the maximum value of \(c\) should be \(\sqrt {18}\).

Then I solved the inequality \(f(-\sqrt{18})\cdot f(\sqrt{18}) <0\) and got 10 integral values for \(m\), which apparently is wrong.

WHYYYYYYYYYY?!

Answer 2

Pretty sure that condition doesn't need any discriminant or stuff. Am I missing something?

Answer 3

I follow everything so far up to the end where you did f(-c)*f(c)<0 Not sure why that would work out.

Answer 4

m+2 +/- sqrt(m^2+2m+4-20m)
-sqrt18<--------------------------- < sqrt 18
2

Answer 5

Intermediate Value Theorem.
I have to make a root lie between -sqrt18 and +sqrt18.

Answer 6

Why not just count all the integers between \[\LARGE -3\sqrt{2} \le m \le 3 \sqrt{2}\]

Answer 7

i didn't get this part
`which apparently is wrong.`

why do you think the count is wrong ?

Answer 8

Because that's the wrong answer, of course.

Answer 9

why

Answer 10

Maybe because this is a parabola and they are specifying "EXACTLY one root"

Answer 11

Are you asking why it is the wrong answer?

Probably because I made a mistake along the way. :|

Answer 12

@Kainui IVT works for that.

Answer 13

It does? I am imagining you having two roots even though you are looking for answers that only give one.

Answer 14

no, im asking why do you think it is "apparantely" the wrong answer..

Answer 15

I don't, but the questionmaker does. lol

Could you guys solve this and confirm the answer as 10?

Answer 16

based on your work, if u get 10 integer solutions then it is the right answer. i don't see anything wrong in ur work or in applying IVT

Answer 17

http://www.wolframalpha.com/input/?i=%2818+-+sqrt%2818%29+%28m%2B2%29+%2B+5m%29+*+%2818+%2B+sqrt%2818%29+%28m%2B2%29+%2B+5m%29+%3C+0

Answer 18

hey explain IVT

Answer 19

Oh wait, you want only one root to lie between -c and c, i think you are not accounting for both roots lying betwee -c and c

Answer 20

That's what I'm saying lol

Answer 21

But if both roots lie between -c and +c, then f(-c) * f(c) > 0. :P

Answer 22

Oh yes you're right right !

Answer 23

ohh i get it cool

Answer 24

I still don't get it. Guess I never learned the IVT

Answer 25

this is a positive parabola

Answer 26

no no it make sesne basically u know hwo at the root its a zero

Answer 27

see if that gif confirms your count parth