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: $$4y^2 - 2 - 9y + 4 = 0$$ Simplify: $$4y^2 - 9y + 2 = 0$$ 3. **Identify the quadratic equation:** It is in the 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:** $$b^2 - 4ac = (-9)^2 - 4 \times 4 \times 2 = 81 - 32 = 49$$ 6. **Find the roots:** $$y = \frac{9 \pm \sqrt{49}}{8} = \frac{9 \pm 7}{8}$$ 7. **Calculate each solution:** - $$y = \frac{9 + 7}{8} = \frac{16}{8} = 2$$ - $$y = \frac{9 - 7}{8} = \frac{2}{8} = \frac{1}{4}$$ 8. **Final answer:** The solutions are $$y = 2$$ and $$y = \frac{1}{4}$$, which corresponds to option b.