1. The problem is to determine which condition holds true if the quadratic equation $ax^2+bx+c=0$ has roots $\ell$ and $\ell$, where $\ell \in \mathbb{R}$. 2. If the roots are equal, then the quadratic has a repeated root $\ell$, meaning the discriminant must be zero. 3. The discriminant $\Delta$ is given by $$\Delta = b^2 - 4ac.$$ For equal roots, $$\Delta = 0,$$ so $$b^2 = 4ac.$$ 4. Dividing both sides of the equation by $4ac$ (assuming $a \neq 0$ and $c \neq 0$), we get $$\frac{b^2}{4ac} = 1.$$ 5. This matches option (d) $\frac{b^2}{4ac} = 1$. 6. Options (a) $a=c$, (b) $c=\ell$, and (c) $b=0$ are not necessarily true for equal roots. Final answer: (d) $\frac{b^2}{4ac} = 1$.