Detemine the value of \( k \) on \( k x^{2}+3 x-9=0 \)

To determine the value of \( k \) for the quadratic equation \( k x^{2}+3 x-9=0 \), we need to ensure that the quadratic has real solutions. This is dictated by the discriminant, \( D \), which for the equation \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \). Here, \( a = k \), \( b = 3 \), and \( c = -9 \). Therefore, the discriminant is: \[ D = 3^2 - 4(k)(-9) = 9 + 36k \] For the quadratic equation to have real solutions, the discriminant must be greater than or equal to zero: \[ 9 + 36k \geq 0 \] Solving for \( k \): \[ 36k \geq -9 \implies k \geq -\frac{1}{4} \] Thus, the value of \( k \) must be greater than or equal to \(-\frac{1}{4}\) for the equation to have real solutions.