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show that e^x-2 = x^3 has exactly two roots

show that e^x-2 = x^3 has exactly two roots by using a graph.
Original post by rakib567
show that e^x-2 = x^3 has exactly two roots by using a graph.


What've you tried doing so far?

If you're having trouble getting started, think about:

- What graph (or graphs) you need to sketch?
- What is meant by a 'root' of an equation?

Note that the question is not asking you to calculate the roots. Just to show that they exist.
Reply 2
Avatar for rakib567
rakib567
OP
15
Original post by Lolgarithms
What've you tried doing so far?

If you're having trouble getting started, think about:

- What graph (or graphs) you need to sketch?
- What is meant by a 'root' of an equation?

Note that the question is not asking you to calculate the roots. Just to show that they exist.


i'm just trying to draw the graph but they do not seem to meet at 2 points
Reply 3
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TeeEm
19
Original post by rakib567
i'm just trying to draw the graph but they do not seem to meet at 2 points


you missed lolgarithm's hint

ex - 2 = x3
ex= x3 + 2
this means sketch 2 graphs and see that these graphs meet in 2 places.

(note that the gradient of ex eventually gets steeper than that of x3)
Reply 4
Avatar for rakib567
rakib567
OP
15
Original post by TeeEm
you missed lolgarithm's hint

ex - 2 = x3
ex= x3 + 2
this means sketch 2 graphs and see that these graphs meet in 2 places.

(note that the gradient of ex eventually gets steeper than that of x3)


so where do they intercept?
Reply 5
Avatar for TeeEm
TeeEm
19
Original post by rakib567
so where do they intercept?


they should intersect in 2 places so that shows the equation has 2 solutions (roots)

where they intersect?

I do not know and the question does not ask for this.
Original post by rakib567
so where do they intercept?


The question doesn't ask you to work out where they intersect. You know that the equation in the question has two roots if you sketch graphs of both sides of the equation and show that they cross in two places. It doesn't matter where they cross, just that they *do* cross means that there are solutions to that equation.

But, as an FYI...

If you try and solve for that using algebra, you should find that you can't do it.

If you take logs of both sides, you'll get

x=ln(x3+2) x = ln (x^3 + 2)

There's not really anywhere that you can go from there to get x on its own. That's kind of the point of the question.
Reply 7
Avatar for rakib567
rakib567
OP
15
Original post by Lolgarithms
The question doesn't ask you to work out where they intersect. You know that the equation in the question has two roots if you sketch graphs of both sides of the equation and show that they cross in two places. It doesn't matter where they cross, just that they *do* cross means that there are solutions to that equation.

But, as an FYI...

If you try and solve for that using algebra, you should find that you can't do it.

If you take logs of both sides, you'll get

x=ln(x3+2) x = ln (x^3 + 2)

There's not really anywhere that you can go from there to get x on its own. That's kind of the point of the question.


Thanks for the explanation but i just wanted to know where the lines meet tbh.
Reply 8
Avatar for Notnek
Notnek
21
Original post by rakib567
Thanks for the explanation but i just wanted to know where the lines meet tbh.

http://www.wolframalpha.com/input/?i=e^x-2+%3D+x^3
Reply 9
Avatar for Arvind54
Arvind54
9
Original post by rakib567
show that e^x-2 = x^3 has exactly two roots by using a graph.


But the graph of e^x and graph of x^3-2 will intersect only at one place??????
Original post by Arvind54
But the graph of e^x and graph of x^3-2 will intersect only at one place??????


if you draw the graph of x^3-e^x+2 then you will see it will cut x-axis at two places.

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