Find without solving?!

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June 24th 2012, 02:13 AM
eskimogenius

Find without solving?!

I came across this question in Cambridge. I tried for ages, and I couldn't get it.

For the equation x^2 = 5x - 8, find the value of a^(1/3) + b^(1/3), where a and b are the roots of the equation, without solving the equation.

Thanks.
June 24th 2012, 02:36 AM
Prove It

Re: Find without solving?!

Why can't you solve the equation?
June 24th 2012, 02:43 AM
eskimogenius

Re: Find without solving?!

Quote:

Originally Posted by Prove It

Why can't you solve the equation?

Mainly because the question says you're not meant to. But if you do solve it, and find values for a and b you'll end up with complex roots and it doesn't look pretty when you take the cube root of that. Essentially you're meant to have the final answer as -1 or something with a root 21 in it. I'll check and post the answers up when I get home.
June 24th 2012, 05:50 AM
HallsofIvy

Re: Find without solving?!

The equation is $x^2- 5x+ 8= 0$ . If a and b are roots of that equation, then $(x- a)(x- b)= x^2- (a+ b)x+ ab= x^2- 5x+ 8$ so we must have a+ b= 5 and ab= 8. Now we can use $a+ b= (a^{1/3}+ b^{1/3})(a^{2/3}+ (ab)^{1/3}+ b^{2/3})$ which, with a+ b= 5, ab= 8 becomes $5= (a^{1/3}+ b^{1/3})(2+ a^{2/3}+ b^{2/3})$ .

Now, look at the fact that [tex](a^{1/3}+ b^{1/3})^2= a^{2/3}+ 2(ab)^{1/3}+ b^{2/3}= 4+ a^{2/3}+ b^{2/3}. That is, $2+ a^{2/3}+ b^{2/3}= -2+ (a^{1/3}+ b^{1/3})^2$ . Putting that into the previous equation, and letting $a^{1/3}+ b^{1/3}= X$ we have $5= X(-2+ X^2)$ or $X^3- 2X- 5= 0$ .
June 25th 2012, 02:57 AM
eskimogenius

Re: Find without solving?!

Thanks mate. I think I got it from there. :)
June 25th 2012, 05:10 AM
BobP

Re: Find without solving?!

We are still not to the bottom of this question.
Sorry HallsofIvy, but you lost a negative sign in your factorisation of $a+b,$ the middle term in the second bracket should be negative.
Following that through I think that the equation at the end should be $X^{3}-6X-5=0.$
This has an obvious solution of $X=-1,$ and the other two are then found to be $2.7913$ and $-1.7913$ (4dp).
I suppose that it is the positive root that we are looking for.