Question

For what values of \( k \) does the quadratic equation \( (k+1)x^2 - 2(k-1)x + 1 = 0 \) have equal roots?

Hint:

Set the discriminant to zero for conditions of equal roots.

Answer 1

For equal roots, the discriminant must be zero:
\[ \Delta = b^2 - 4ac = 0. \] \[ [ -2(k-1)]^2 - 4(k+1)(1) = 0. \] \[ 4(k-1)^2 - 4(k+1) = 0. \] \[ (k-1)^2 - (k+1) = 0. \] Expanding:
\[ k^2 - 2k + 1 - k - 1 = 0. \] \[ k^2 - 3k = 0. \] \[ k(k-3) = 0. \] \[ k = 0 \text{ or } k = 3. \]