1. **State the problem:** Find the discriminant of the quadratic equation $$1 - 5x^2 - 2x = 0$$ and determine what the discriminant tells us about the number of real solutions. 2. **Rewrite the equation in standard form:** The standard form of a quadratic equation is $$ax^2 + bx + c = 0$$. Rearranging, we get: $$-5x^2 - 2x + 1 = 0$$ Here, $$a = -5$$, $$b = -2$$, and $$c = 1$$. 3. **Recall the discriminant formula:** $$\Delta = b^2 - 4ac$$ The discriminant tells us: - If $$\Delta > 0$$, there are two distinct real solutions. - If $$\Delta = 0$$, there is exactly one real solution. - If $$\Delta < 0$$, there are no real solutions (solutions are complex). 4. **Calculate the discriminant:** $$\Delta = (-2)^2 - 4 \times (-5) \times 1 = 4 + 20 = 24$$ 5. **Interpret the result:** Since $$\Delta = 24 > 0$$, the quadratic equation has two distinct real solutions. **Final answer:** The discriminant is equal to 24, which means the equation has two real number solutions.