Find the value of m for which the quadratic equation (m-1)x² + 2(m-1)x + 3 = 0 has equal roots — JEE Mathematics

For a quadratic equation $Ax^2 + Bx + C = 0$ to have equal roots, its discriminant $D = B^2 - 4AC$ must equal $0$, and $A \ne 0$.
Here, $A = m-1$, $B = 2(m-1)$, and $C = 3$.

First, $m - 1 \ne 0 \implies m \ne 1$.
Now, set $D = 0$:

$$[2(m-1)]^2 - 4(m-1)(3) = 0$$

$$4(m-1)^2 - 12(m-1) = 0$$

$$4(m-1)[(m-1) - 3] = 0$$

$$4(m-1)(m-4) = 0$$

This gives $m = 1$ or $m = 4$.
Since $m \ne 1$ for the equation to remain quadratic, we discard $m = 1$.

Answer: $m = 4$