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Difficult cubic equation.

By using a substitution to remove the term in

x^2

, or otherwise, find the single real solution to the equation

\displaystyle 16x^3+96x^2+180x+99=0

Reply 1

the bear

how far have you got ?

Reply 2

Ano123

Original post by the bear

how far have you got ?

Used substitution

x=y-2

to get it down to

16y^3-12y-5=0

this should help:

Original post by Mathemagicien

Remember cos 3x = 4 cos^3x - 3cosx

I'm not sure if that helps here.
I'd get

\cos 3\alpha = 5/4

if I just did it like that. :s-smilie:

(edited 7 years ago)

Reply 5

Ano123

Original post by Mathemagicien

e^{3i \alpha }+e^{-3i \alpha } = 5/2

which is a quadratic in

e^{3i \alpha }

which gives you

e^{3i \alpha }=2,1/2

take logs, we get

\alpha = +- \dfrac{log 2}{3i} = -+ \dfrac{i log 2}{3}

and you can find cos

\alpha

from that

I get cos

\alpha = 2^{-2/3}+2^{-4/3}

Not very nice looking solutions though are they? What have you done to get

\cos \alpha

from there?
Oh, I see.

(edited 7 years ago)

Reply 6

Ano123

Original post by Mathemagicien

So use cos

x = \dfrac{e^{ix} + e^{-ix}}{2}

\dfrac{e^{3i \alpha }+e^{-3i \alpha } }{2} = 5/4

which is a quadratic in

e^{3i \alpha }

which gives you

e^{3i \alpha }=2,1/2

take logs, we get

\alpha = +- \dfrac{log 2}{3i}

and you can find cos

\alpha

(and thus y and thus x) from that

I get cos

\alpha = 2^{-2/3}+2^{-4/3}

(which satisfies our equation)

What does the

\mathbf{+ -}

mean?

Reply 7

RDKGames

Study Forum Helper

Original post by Mathemagicien

Plus or minus (don't know the latex for it)

\pm

Reply 8

Ano123

Original post by Mathemagicien

Plus or minus (don't know the latex for it)

Thought so. \pm - code for it.
I'll tell you I have got the solution to this equation but I used

y=\cosh \alpha

. It's slightly easier doing this but I actually like your method a lot. Thanks!

(edited 7 years ago)

Reply 9

Ano123

Original post by Mathemagicien

You could also use y=cosh t, cosh 3t = 4 cosh^3 t - 3 cosh t, its equivalent

That's what I meant I used cosh not sinh.

Reply 10

Randall13

Exams are over. Chill the **** out.

Reply 11

RDKGames

Study Forum Helper

Original post by Randall13

Exams are over. Chill the **** out.

Why?

Reply 12

ODES_PDES

Original post by Randall13

Exams are over. Chill the **** out.

this!

Reply 13

Ano123

Original post by Randall13

Exams are over. Chill the **** out.

I don't do this for exams. Exams are over and I'm bored so may as well do some more maths, what could I be doing that's more interesting than that?

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