1. **State the problem:** Find the solutions of the quadratic equation $$5x^2 - 2x - 9 = 0$$ where $$x \in \mathbb{R}$$, correct to 2 decimal places. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=5$$, $$b=-2$$, and $$c=-9$$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 5 \times (-9) = 4 + 180 = 184$$ 4. **Evaluate the square root of the discriminant:** $$\sqrt{184} \approx 13.5647$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-2) \pm 13.5647}{2 \times 5} = \frac{2 \pm 13.5647}{10}$$ 6. **Find the two solutions:** - For the plus sign: $$x_1 = \frac{2 + 13.5647}{10} = \frac{15.5647}{10} = 1.55647 \approx 1.56$$ - For the minus sign: $$x_2 = \frac{2 - 13.5647}{10} = \frac{-11.5647}{10} = -1.15647 \approx -1.16$$ 7. **Final answer:** The solutions to the equation $$5x^2 - 2x - 9 = 0$$ are $$x \approx 1.56$$ and $$x \approx -1.16$$, correct to 2 decimal places.