Question

Locate between consecutive integers each real root of each equation...

Answer 1

\[x^3-x^2-5=0\]

Answer 2

@terenzreignz

Answer 3

Okay... here we go... looks like trouble... how were you taught to solve equations like this?

Answer 4

a bunch of different ways.... but I don't know which way to go.

Answer 5

Well first of all, let me just say that this has no rational roots...

Answer 6

So, I guess we're just supposed to find a pair of consecutive integers that contain a root between them.

Answer 7

I believe so...

Answer 8

Well, it's really simple, see.
All you have to do is find a pair of consecutive integers, say, n and n+1
such that the signs of f(n) and f(n+1) are different.

Answer 9

Intermediate value theorem, as I recall.

Answer 10

haha sounds hard... so what would be my first step in this?

Answer 11

Well, try zero.

Answer 12

I'm really barely getting through these root sections of my book so bare with me lol... but okay do you want me to plug in 0?

Answer 13

First of all, let's define
\[\Large f(\color{red}x) = \color{red}x^3 - \color{red}x^2 -5\]
And we want $$\Large f(\color{red}x) = 0$$

So, try f(0) what do you get?

Answer 14

well I get \[x^3-x^2-5=0\]

Answer 15

since f(x)=0

Answer 16

if I plug in 0 for x, I get -5

Answer 17

Good. It's negative, right?
Now, keep plugging in consecutive integers.. (1,2,3... etc) UNTIL it changes sign.

Let me know when you're done.

Answer 18

okay hold on...

Answer 19

when I plugged in 3 I got 13

Answer 20

what about 2?

Answer 21

-1

Answer 22

Okay, so it changed sign from 2 to 3, right?
Means there's a root between 2 and 3 ^_^

Answer 23

wow that's easier the I thought!!! thank you soooo much!

Answer 24

Hold on there...

Answer 25

There might be other roots. But from 3 onwards, that x^3 is going to be so big, it's going to make the -x^2 - 5 rather insignificant, right?

Answer 26

IE, it's going to be positive (and not change anymore) from x = 3 onwards.
Understood?

Answer 27

hmm.. is there an example support this?

Answer 28

Well, try plugging in 4 onwards.
Observe that the value just explodes upwards (IE, gets bigger and bigger)
And if it gets bigger and bigger, it's not going to be zero, and we want it to be zero.

The reason we're sure that there's a zero somewhere between 2 and 3 is because at x = 2, the function was negative, and at x = 3, the function was positive, so in going from negative to positive, the curve must have passed through 0 at some point.

Answer 29

what about between x=2 and x=4 , couldn't any number between x=2 and x=3 or over be the answer.

Answer 30

lol
since there's a root between 2 and 3, then surely, there's a root between 2 and 4, and that's precisely the root between 2 and 3.

haha
didn't you see that nything between 2 and 3 is also between 2 and 4?

Answer 31

cause there is a zero between x=2 and x=4 right?

Answer 32

There is, and it's the same zero between 2 and 3.
But remember you're asked for a pair of *consecutive* integers

Answer 33

ahhh now I get it! yes 2 and 3 are consecutive!

Answer 34

thanks for the help!

Answer 35

No problem.
By the way, you really SHOULD try going from zero the left, this time (IE, -1, -2, -3)
Just to verify that there will be no sign-change in that direction too ^_^

Answer 36

okay will do that. :D

Answer 37

Graph.
Just so you can clearly see that there is only one root.
https://www.google.co.uk/#q=x%5E3+-+x%5E2+-+5