1. **State the problem:** Solve the quadratic equation $$9x^2 - 7x - 23 = 0$$ using the quadratic formula. 2. **Quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = 9$$, $$b = -7$$, and $$c = -23$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-7)^2 - 4 \times 9 \times (-23) = 49 + 828 = 877$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-7) \pm \sqrt{877}}{2 \times 9} = \frac{7 \pm \sqrt{877}}{18}$$ 6. **Simplify the expression:** Since 877 is not a perfect square, the solutions are irrational. 7. **Final solutions:** $$x_1 = \frac{7 + \sqrt{877}}{18}$$ $$x_2 = \frac{7 - \sqrt{877}}{18}$$ **Solution type:** Irrational **Solved by:** Quadratic Formula