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 find the values of \( k \) for the given quadratic equation \( x^{2}-5x-k=0 \), we can utilize the discriminant \( D = b^2 - 4ac \), where \( a = 1 \), \( b = -5 \), and \( c = -k \). For equal roots, the discriminant must be zero: \[ D = (-5)^2 - 4(1)(-k) = 25 + 4k = 0 \] Solving for \( k \) gives: \[ k = -\frac{25}{4} \] For real roots, the discriminant must be non-negative: \[ D \geq 0 \Rightarrow 25 + 4k \geq 0 \Rightarrow 4k \geq -25 \Rightarrow k \geq -\frac{25}{4} \] For non-real roots, the discriminant must be negative: \[ D < 0 \Rightarrow 25 + 4k < 0 \Rightarrow 4k < -25 \Rightarrow k < -\frac{25}{4} \]