1. **State the problem:** Solve for $y$ in the equation $$4y^2 - 2 = 9y - 4$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$4y^2 - 2 - 9y + 4 = 0$$ which simplifies to $$4y^2 - 9y + 2 = 0$$. 3. **Identify the quadratic form:** The equation is now in standard quadratic form: $$ay^2 + by + c = 0$$ where $a=4$, $b=-9$, and $c=2$. 4. **Use the quadratic formula:** $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-9)^2 - 4 \times 4 \times 2 = 81 - 32 = 49$$. 6. **Find the roots:** $$y = \frac{-(-9) \pm \sqrt{49}}{2 \times 4} = \frac{9 \pm 7}{8}$$. 7. **Evaluate each solution:** - For the plus sign: $$y = \frac{9 + 7}{8} = \frac{16}{8} = 2$$ - For the minus sign: $$y = \frac{9 - 7}{8} = \frac{2}{8} = \frac{1}{4}$$ **Final answer:** $$y = 2 \quad \text{or} \quad y = \frac{1}{4}$$