how to solve this algebric equation

gyanshrestha · 2007-11-08 18:04:28

what is the value of a^4-a^3+a^2+2 if a^2+2=2a

Jai Ganesh · 2007-11-08 21:21:20

gyanshreshta,
this is polynomial (actually, a monomial since only one variable is there in it)of degree 4, that is the highest power is 4.
You have not stated a^4-a^3+a^2+2=0.
Since = 0 is not given, it is only a ploynomial and not an equation.
It has already been given a^2+2=2a.
Substitute it in the original polynomial.
a^4-a^3+a^2+2=a^4-a^3+2a.
To factorise this, take the a outside the bracket. You get
a (a^3-a^2+2).

PS:- If you want to solve a cubic equation (degree 3) or quartic equation (degree 4) I think there's no prescribed method. I haven't come across any. I would do it by 'Trial and success' method.

mathsyperson · 2007-11-09 00:07:45

You can use that identity many times to reduce the equation's order.
For example, multiplying both sides by a² gives that a^4 + 2a^2 = 2a^3.
Therefore, a^4-a^3+a^2+2 = a^3-a^2+2.

Similarly, a^3 + 2a = 2a^2, and that means that a^3-a^2+2 = a^2-2a+2.

Then, because a^2+2 = 2a, that gives a^2+2-2a = 0.

TheDude · 2007-11-09 01:12:53

I think you guys are making this too hard. All he has to do is solve for

using the equation given to him:

Then once you find

you just substitute it back into the first polynomial to get it's value.

mathsyperson · 2007-11-09 01:44:42

Heh, fair point.

Although looking at it, that would return a complex value of a, so it wouldn't be quite as trivial as you suggest.

TheDude · 2007-11-09 01:50:23

I didn't even think about that, but you're right. I hope he doesn't mind finding 4th powers of complex numbers.

mathsyperson · 2007-11-09 05:53:39

It's not actually too bad. Solving the quadratic gives a = 1±i.

Therefore, a² = ±2i and so a^4 = -4.
a³ = ±2i(1±i) = ±2i - 2.

Plugging all of that in gives that a^4-a^3+a^2+2 = -4 -(±2i - 2) ±2i + 2
= (-4+2+2) + (±2 - ±2)i = 0.

Two methods, same answer. More or less equally strenuous as well.

gyanshrestha · 2007-11-09 19:40:03

thanks mathsyperson.
i myself finally did it in another way.
ie
=^2(a^2-2a+2)+a(a^2-2a+2)+1(a^2-2a+2)
=a^2+a+1)(a^2-2a+2)
=a^2+a+1)*0
=0

Math Is Fun Forum

#1 2007-11-08 18:04:28

how to solve this algebric equation

#2 2007-11-08 21:21:20

Re: how to solve this algebric equation

#3 2007-11-09 00:07:45

Re: how to solve this algebric equation

#4 2007-11-09 01:12:53

Re: how to solve this algebric equation

#5 2007-11-09 01:44:42

Re: how to solve this algebric equation

#6 2007-11-09 01:50:23

Re: how to solve this algebric equation

#7 2007-11-09 05:53:39

Re: how to solve this algebric equation

#8 2007-11-09 19:40:03

Re: how to solve this algebric equation

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