Help me solve an algebra problem

[wcz]

could anyone helps me to solve the following question

a^2 + a - 1 = a^3/2

what is a?
is this equation can be solved?
Thanks

 

[comm_eng]

Re: algebra problem

try to solve it numerically

here is a method :

by trial and erreor

a(new)=a(old)^(3/2)-a(old)^2+1

and put an inital value for a in the right hand side (say =0.5) and get the new value for a(new) .. sub with this value again in the right hand side... and so on till u statisfied with the accuracy...

if u find the solution diverts,, try to change the a(new)

eg.

a(new)=(a(old)^2+a(old)-1)^(2/3)


there r many other numerical ways to solve such problems..

another way is to use the fx2 algebra casio calculator :b

 

[eng_Semi]

Re: algebra problem

I didn't think that he mentioned a(new) and a(old).

If we assume that we have ONLY one "a":
Squaring both sides:
[a^(3/2)]^2 = (a^2 + a -1)^2
a^3 = a^4 + a^3 - a^2 + a^3 + a^2 - a - a^2 - a + 1
a^4 + a^3 - a^2 - 2a + 1 = 0
I think that this form of equation can be factorized, but I didn't remember how :'( :'(

 

[Kral]

Re: algebra problem

eng_Semi is on the right track! You don't need to factor the equation. Any algebra book will have the formulas for determining a closed form solution. Basically, you resolve the 4th order equation into a cubic equation with a change of variable. You then solve the cubic using Cardan's formulas.
.
See, for example "Handbook of Mathematical Tables and Fourmulas" by Burington, or the CRC Math tables (Not the exact title, but I don't have my copy handy).
.
Regards,
Kral

 

[steve10]

Re: algebra problem

Set x=a^(1/2). Then your equation becomes
x^4 - x^3 + x^2 - 1=0
which can be factored as
(x - 1)(x^3 + x + 1) = 0

Therefore, you have a solution x=1, which means a=1.

As for x^3 + x + 1 = 0. Algebra tells you that any rational solution has to be a factor of the const term which is 1 in this case. Therefore, you have only two choice, 1 or -1. Direct substitutions show that none of them is the solution of x^3 + x + 1 = 0. Thus, you are left out with only irrational solutions.

Further analysis can be done as follows. Set
f(x)=x^3 + x + 1
Take derivative
f'(x)=3*x^2+1 >0
which means that f(x) is an increasing function and, therefore, has only one real solution. The other two are complex solutions.

 

[whynot910]

Re: algebra problem

an algebra polynomial equation containing degree 4 or less can be solved by a formula in terms of radicals.

 

[net]

algebra problem

hi, I Know that many people have different methods and want to solveit by algebra but the answer can be found easily by just setting a graphic, a^2 + a - 1 = a^3/2 change to a^2 + a - a^3/2 = 1, you will seee that the only number for "a" to acomplish these constraints will be the number one, what you can do also is to derive it

 

[Dr_mas]

Re: algebra problem

let f(a) = a^2 + a - 1 - a^3/2 and plot the function by matlab or any equation graph program.
read the points of intersection between f and a - axis » (f = 0) then it will give your answer .
if there is no inersection so there is no answer to this eguation

mas