Question

solve for x from the given equation. ​

Answer 1

Answer:

this margin is not large enough to contain.

Relax, Just kidding!

Here goes the solution:

Assuming you want a closed form solution and not an approximation, this problem is not really hard to solve. Knowing a trick or two will make your life a lot easier.

You can check for yourself that x2n+1x2n+2=(xn+1xn)2x2n+1x2n+2=(xn+1xn)2

Using this simple trick, I’m going to add 22 to both the sides of the equation and factor it out completely as a square. So now we have,

X4+2+1X4=20⟹(X2+1X2)=20−−√X4+2+1X4=20⟹(X2+1X2)=20 .

If you see the pattern, then that’s awesome we’re good to go. Hopefully, one more iteration and we shall find an easier expression to work with.

Now,

X2+2+1X2=2+20−−√⟹(X+1X)2=2+20−−√⟹X+1X=2+20−−√−−−−−−−√.X2+2+1X2=2+20⟹(X+1X)2=2+20⟹X+1X=2+20.

Finally, we now have something we can rather solve easily (in fact, by using quadratic formula for roots).

So X+1X=2+20−−√−−−−−−−√⟹X2+1X=2+20−−√−−−−−−−√⟹X2−(2+20−−√−−−−−−−√)X+1=0X+1X=2+20⟹X2+1X=2+20⟹X2−(2+20)X+1=0

Using the quadratic formula for roots of this quadratic, we have X=−b±b2−4ac−−−−−−−√2aX=−b±b2−4ac2a

where a=1,b=2+20−−√−−−−−−−√,c=1a=1,b=2+20,c=1

Finally we have, X=−2+20−−√−−−−−−−√±2+20−−√−4−−−−−−−−−−√2X=−2+20±2+20−42

∴X=−2+20−−√−−−−−−−√±20−−√−2−−−−−−−√2∴X=−2+20±20−22 .

Hence, X1=−2+20−−√−−−−−−−√+20−−√−2−−−−−−−√2X1=−2+20+20−22 and X2=−2+20−−√−−−−−−−√−20−−√−2−−−−−−−√2X2=−2+20−20−22 are the two potential solutions to the equation X4+1X4=18X4+1X4=18 .

You can check it for yourself , the accuracy of this method by approximating this surd and comparing with the other answers on this thread.

Hope it helps!

In case you’re curious to solve similar problems, I’d like to suggest you this problem we were assigned from one of the problem solving sessions I’ve attended. If x−−√3+1x−−√3=3,x3+1x3=3, find with proof, the value of x3+1x3x3+1x3 .

Have a fun time solving this one!

Answer 2

Refer the attachment for your answer.

hope it helps