China | Math Olympiad Problem | A Nice Algebra Challenge 

Thank you for this - most people would give up quickly because, unless you know the trick of adding a "1" to both sides to get the perfect square, you come to a fourth degree equation! I didn't manage to make the perfect square so I gave up. Certainly it's a nice algebra challenge.

Nice example with +1 trick!🎉

Good. But time can be reduced furher.

Is your mic turned up too much? There are a lot of clipping on the audio.

If ones don't kno😮w the '+1' trick in the lecturer's, following way can be another way. Letting x+1 = y and just expending the given, results in y^4 -2y^3 - y^2 - 2y + 1 = 0 and deviding by y^2 gives (y^2 + 1/y^2) - 2(y + 1/y) - 1 = 0, which can be (y + 1/y)^2 - 2(y +1/y) - 3 = 0. That is a quad. eq. in terms of (y + 1/y) Let again (y + 1/y) = t, then t^2 - 2t -3 = 0; (t+1)(t-3) = 0. Thus t = (y + 1/y) = -1 or 3. Solving for y and then finally for x will end up with the same answers.

asnwer=1x

Since X^2 and X^2 /(X+1)^2 are both square, we subtract and add the twice of first by second one to the left of the equation: (X - X/(X+1))^2 + 2X^2?(X+1) = 3 (X- X/X+1)^2 + 2X^2/(X+1) = 3 ((X^2+X-X)/(X+1))^2 + 2X^2/(X+1) = 3 (X^2/(X+1))^2 + 2X^2/(X+1) = 3 X^2/(X+1) = t t^2 +2t - 3 = 0 So: t = - 3, t = 1 X^2/(X+1) =1 So: X^2 - X - 1 = 0 X= 1/2 +, - Sqrt (5) X^2/(X+1) = - 3 So: X^2+3X+3 = 0 X = - 3/2 +, - Sqrt(3). i As you see this is easier approach to solve the above equation.

One solution is the golden ratio.

Why -2; V(4)=2 but not -2. This is a conceptual error