1. The problem statement, all variables and given/known data Find the points on the lemniscate: 2( x^2 + y^2 )^2 = 25( x^2 - y^2 ) where the tangent is horizontal 2. Relevant equations Horizontal tangent: y' = 0 3. The attempt at a solution I got the correct gradient of y' = [ 50x - 8x^3 - 8y^2 ] / [ (8x^2)y + 50y + 8y^3 ], and then solved for y = + ( -8x^3 + 50x)^1/2 ), after which I subbed this y into the original equation and tried to solve for x, whereby I reached a dead end. The last line I got to was complicated when solving for x: 128x^6 - 32x^5 + 2x^4 - 1800x^23 + 175x^2 + 1250x = 0.