Question: Use the quadratic formula to solve for $x$.
$4x^2 - 9x + 3 = 0$
(If there is more than one solution, separate them with commas.)
1. **State the problem:** Solve the quadratic equation $$4x^2 - 9x + 3 = 0$$ using the quadratic formula.
2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions for $$x$$ are given by:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
3. **Identify coefficients:** Here, $$a = 4$$, $$b = -9$$, and $$c = 3$$.
4. **Calculate the discriminant:**
$$\Delta = b^2 - 4ac = (-9)^2 - 4 \times 4 \times 3 = 81 - 48 = 33$$
5. **Apply the quadratic formula:**
$$x = \frac{-(-9) \pm \sqrt{33}}{2 \times 4} = \frac{9 \pm \sqrt{33}}{8}$$
6. **Final solutions:**
$$x_1 = \frac{9 + \sqrt{33}}{8}, \quad x_2 = \frac{9 - \sqrt{33}}{8}$$
These are the two solutions for $$x$$.