olve using the appropriate (a) \( \frac{2}{3} x^{2}-\frac{5}{2} x+\frac{1}{4}=0 \)

We start with the equation \[ \frac{2}{3}x^2-\frac{5}{2}x+\frac{1}{4}=0. \] **Step 1. Clear the fractions** The denominators are 3, 2, and 4. The least common multiple is 12. Multiply the entire equation by 12: \[ 12\left(\frac{2}{3}x^2\right)-12\left(\frac{5}{2}x\right)+12\left(\frac{1}{4}\right)=0. \] Calculate each term: \[ 12\left(\frac{2}{3}x^2\right)=8x^2,\quad 12\left(\frac{5}{2}x\right)=30x,\quad 12\left(\frac{1}{4}\right)=3. \] Thus, the equation becomes \[ 8x^2-30x+3=0. \] **Step 2. Use the quadratic formula** The quadratic formula is \[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}. \] For our equation, \(a=8\), \(b=-30\), and \(c=3\). Substitute these values: \[ x=\frac{-(-30)\pm\sqrt{(-30)^2-4\cdot8\cdot3}}{2\cdot8}=\frac{30\pm\sqrt{900-96}}{16}. \] **Step 3. Simplify the discriminant** Compute the discriminant: \[ 900-96=804. \] We simplify \(\sqrt{804}\): \[ 804=4\cdot201,\quad \sqrt{804}=\sqrt{4\cdot201}=2\sqrt{201}. \] **Step 4. Write the final solution** Substitute back into the formula: \[ x=\frac{30\pm2\sqrt{201}}{16}. \] This expression can be simplified by factoring out a 2 in the numerator: \[ x=\frac{2(15\pm\sqrt{201})}{16}=\frac{15\pm\sqrt{201}}{8}. \] Thus, the solutions to the equation are \[ x=\frac{15+\sqrt{201}}{8}\quad \text{and}\quad x=\frac{15-\sqrt{201}}{8}. \]