Solve the equation. Express numbers in exact simplest form. \[ 4 d^{3}-15 d^{2}+9 d=0 \]

To solve the equation \( 4d^{3}-15d^{2}+9d=0 \), we first factor out the common term \( d \): \[ d(4d^{2}-15d+9)=0 \] This gives us one solution immediately: \[ d=0 \] Next, we need to solve the quadratic equation \( 4d^{2}-15d+9=0 \). We can apply the quadratic formula: \[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 4 \), \( b = -15 \), and \( c = 9 \). Calculating the discriminant: \[ b^2 - 4ac = (-15)^2 - 4 \cdot 4 \cdot 9 = 225 - 144 = 81 \] Now using the quadratic formula: \[ d = \frac{15 \pm \sqrt{81}}{2 \cdot 4} = \frac{15 \pm 9}{8} \] Calculating the two potential solutions: 1. For the positive case: \[ d = \frac{15 + 9}{8} = \frac{24}{8} = 3 \] 2. For the negative case: \[ d = \frac{15 - 9}{8} = \frac{6}{8} = \frac{3}{4} \] Thus, the solutions to the equation \( 4d^{3}-15d^{2}+9d=0 \) are: \[ d = 0, \quad d = 3, \quad d = \frac{3}{4} \]