Math Is Fun Forum

PureTree

Member

Registered: 2007-09-24

Posts: 0

Please help me solve that exercise!

3(x²+4x)² - 2(x+2)² = 57

pi man

Member

Registered: 2006-07-06

Posts: 251

Re: Please help me solve that exercise!

X=1 will work

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Please help me solve that exercise!

3(x²+4x)² - 2(x+2)² =  3(x^4 + 8x³ + 16x²) - 2(x² + 4x + 4) = 3x^4 + 24x³ + 48x² - 2x² - 8x - 8 = 3x^4 + 24x³ + 46x² - 8x - 8 = 57

3x^4 + 24x³ + 46x² - 8x - 65 = 0

Which is a quartic equation that you can read about here.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Please help me solve that exercise!

Using Pi man's x=1 and Synthetic Division I factored it down to:

(x-1)(3x³ + 27x² + 73x + 65) = 0

I'm curious: Pi Man, what method did you use to find that value?

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Please help me solve that exercise!

x = −5 is another solution.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Please help me solve that exercise!

Using Pi man's x=1 and Synthetic Division I factored it down to:

(x-1)(3x³ + 27x² + 73x + 65) = 0

I'm curious: Pi Man, what method did you use to find that value?

Guess-and-check is a pretty common method of finding factors... although he could have used some new top-secret technique

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Please help me solve that exercise!

x = −5 is another solution.

Yeah, I saw that graphically, but I was hoping someone would come along and answer it analytically.

By the way, the other two roots are complex.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Please help me solve that exercise!

Okay, here’s a quick way to solve everything. Let X = (x+2)[sup]2[/sup]. Then

Must I do everything around here all the time?

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Please help me solve that exercise!

Must I do everything around here all the time?

Oh come on, We pretty much had that one under control before you got involved.

Awesome answer though.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Please help me solve that exercise!

Well, look closely at the original equation. There must be a reason why it was written that way instead of as a straightforward quartic equation. It must be so as to make it clear that a substitution is meant to be involved – so one should be looking around for a substitution instead of trying to solve the quartic directly.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Please help me solve that exercise!

Well, look closely at the original equation. There must be a reason why it was written that way instead of as a straightforward quartic equation. It must be so as to make it clear that a substitution is meant to be involved – so one should be looking around for a substitution instead of trying to solve the quartic directly.

I understand that, and agree, but these sorts of problems really upset me.

I know they shouldn't, and I know it's irrational, but I kind of feel like problems like these are gimmicky. I wonder how many times in a real-life situation will you come across a problem like that?

Your answer has a certain beauty to it, no doubt. It almost dances as it comes to its conclusion, (as G.H. Hardy said, "The mathematician's patterns, like the painter's or the poet's, must be beautiful"") But I feel like approaching like I did, has more value. I think in real life when you come across an equation very rarely will it be able to be solved with a nifty substitution.

Then again (and I'll contradict myself before someone else does), if I felt that way, why not just graph it and get a "good enough" approximation? Approximations are all you really need in real life.

I guess that's the difference between "Pure Mathematics" and "Applied Mathematics."

Or maybe I'm just jealous that your answer was cooler than mine.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.