1. **State the problem:** We need to estimate the solutions to the equation $$10x - 5x^2 - 1 = 1.2$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$10x - 5x^2 - 1 - 1.2 = 0$$ which simplifies to $$10x - 5x^2 - 2.2 = 0$$. 3. **Rewrite in standard quadratic form:** $$-5x^2 + 10x - 2.2 = 0$$ or equivalently $$5x^2 - 10x + 2.2 = 0$$ (multiplying both sides by -1). 4. **Identify coefficients:** $$a = 5, \quad b = -10, \quad c = 2.2$$. 5. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 6. **Calculate the discriminant:** $$b^2 - 4ac = (-10)^2 - 4 \times 5 \times 2.2 = 100 - 44 = 56$$. 7. **Calculate the square root:** $$\sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14} \approx 7.483$$. 8. **Calculate the two solutions:** $$x = \frac{-(-10) \pm 7.483}{2 \times 5} = \frac{10 \pm 7.483}{10}$$ 9. **First solution:** $$x_1 = \frac{10 + 7.483}{10} = \frac{17.483}{10} = 1.7483$$ 10. **Second solution:** $$x_2 = \frac{10 - 7.483}{10} = \frac{2.517}{10} = 0.2517$$ **Final answer:** The estimated solutions to the equation are approximately $$x \approx 1.75$$ and $$x \approx 0.25$$.