A very nice olympiad question | How to solve 4x^2/(1-sqrt{2x-+1})^2=2x+9 | Algebra |

See the way I breakdown the solution of this question. There is a lot you can learn from this video.
How to solve 4x^2/(1-sqrt{2x-+1})^2=2x+9
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[@robertliu3176]

Let t=sqrt(2x+1), then 2x=t^2-1. Then numerator of LHS =(2x)^2=(t^2-1)^2=(t+1)^2(t-1)^2, denominator of LHS = (1-t)^2=(t-1)^2. Reducing (t-1)^2, the LHS = (t+1)^2=t^2+2t+1. The RHS = t^2-1+9=t^2+8. Eliminating t^2 from both sides, we get t= 7/2. Then x=(t^2-1)/2=45/8.

[@user-dq2tf2lx4w]

I get it

[@SpencersAcademy]

I'm glad you did.

[@user-kp2rd5qv8g]

Write t= sqrt(2x+1). Then the given equation becomes 2t^3-11t^2+16t-7=0, which has the solutions t=1,1,7/2. t=1 is not a valid solution. t=7/2 yields x=45/8, which is the required solution.

[@SpencersAcademy]

Fantastic 👏