Use the discriminant to tell whether the solutions of this equation are real or complex \[ 4 x^{2}-3 x+9=0 \]

To determine the nature of the solutions for the equation \( 4x^2 - 3x + 9 = 0 \), we can use the discriminant, which is given by the formula \( D = b^2 - 4ac \). Here, \( a = 4 \), \( b = -3 \), and \( c = 9 \). Calculating the discriminant gives us: \[ D = (-3)^2 - 4 \cdot 4 \cdot 9 = 9 - 144 = -135. \] Since the discriminant \( D \) is negative, this indicates that the solutions to the equation are complex numbers. When dealing with quadratic equations, complex solutions typically appear as conjugate pairs. In this case, the solutions can be expressed as \( \frac{-b \pm \sqrt{D}}{2a} \). Given \( D < 0 \), you will have terms containing the imaginary unit \( i \), which symbolizes the 'imaginary' part of the solutions, revealing the interesting interplay between real and complex numbers in mathematics!