Fractional equation with no real solutions

Text

Solve the following equation: $\frac{2x}{(x+1)^2} = 1+\frac{x}{x + 1}.$

Solution

Existence condition: $x\neq -1$. Multiply by $(x+1)^2$: $2x = (x+1)^2 + x(x+1) = (x+1)\big[(x+1)+x\big] = (x+1)(2x+1) = 2x^2+3x+1.$ Hence $2x^2 + x + 1 = 0$, with discriminant $\Delta = 1 - 8 = -7 < 0.$ The discriminant is negative: the equation has no real solutions. $\boxed{\text{No real solution}\ (\Delta<0)}$