2. For which value(s) of \( k \) will the equation \( x^{2}-5 x-k=0 \) have: a) equal roots b) real roots c) non-real roots

To determine the values of \( k \) for different types of roots in the quadratic equation \( x^{2}-5x-k=0 \), we need to use the discriminant formula, which is given by \( D = b^2 - 4ac \). Here, \( a = 1 \), \( b = -5 \), and \( c = -k \). For equal roots, the discriminant must be zero: Setting \( D = 0 \), we get: \[ (-5)^2 - 4(1)(-k) = 0 \implies 25 + 4k = 0 \implies k = -\frac{25}{4}. \] For real roots, the discriminant must be greater than or equal to zero: Setting \( D \geq 0 \), we get: \[ 25 + 4k \geq 0 \implies k \geq -\frac{25}{4}. \] For non-real roots, the discriminant must be less than zero: Setting \( D < 0 \), we get: \[ 25 + 4k < 0 \implies k < -\frac{25}{4}. \]