1. **Stating the problem:** We are given the quadratic equation in $x$: $$3x^2 - 2(2a + b)x + 2a^2 + 2ab + b^2 - 1 = 0,$$ with conditions: (1) $a^2 + ab + b^2 \leq \frac{3}{2}$ and (2) $a + b \leq \sqrt{2}$. 2. **Understanding the problem:** We want to analyze this quadratic equation and the given inequalities involving $a$ and $b$. The equation is quadratic in $x$ with coefficients depending on $a$ and $b$. 3. **Formula used:** For a quadratic equation $Ax^2 + Bx + C = 0$, the discriminant is: $$\Delta = B^2 - 4AC,$$ which determines the nature of roots. 4. **Identify coefficients:** $$A = 3,$$ $$B = -2(2a + b) = -4a - 2b,$$ $$C = 2a^2 + 2ab + b^2 - 1.$$ 5. **Calculate the discriminant:** $$\Delta = B^2 - 4AC = (-4a - 2b)^2 - 4 \cdot 3 \cdot (2a^2 + 2ab + b^2 - 1).$$ 6. **Simplify the discriminant:** $$(-4a - 2b)^2 = 16a^2 + 16ab + 4b^2,$$ $$4 \cdot 3 \cdot (2a^2 + 2ab + b^2 - 1) = 12(2a^2 + 2ab + b^2 - 1) = 24a^2 + 24ab + 12b^2 - 12.$$ So, $$\Delta = (16a^2 + 16ab + 4b^2) - (24a^2 + 24ab + 12b^2 - 12) = 16a^2 + 16ab + 4b^2 - 24a^2 - 24ab - 12b^2 + 12.$$ 7. **Combine like terms:** $$\Delta = (16a^2 - 24a^2) + (16ab - 24ab) + (4b^2 - 12b^2) + 12 = -8a^2 - 8ab - 8b^2 + 12.$$ 8. **Factor out -8:** $$\Delta = -8(a^2 + ab + b^2) + 12.$$ 9. **Use inequality (1):** Since $a^2 + ab + b^2 \leq \frac{3}{2}$, $$\Delta \geq -8 \cdot \frac{3}{2} + 12 = -12 + 12 = 0.$$ This means the discriminant is non-negative, so the quadratic has real roots. 10. **Interpret inequality (2):** $a + b \leq \sqrt{2}$ restricts the sum of $a$ and $b$. 11. **Summary:** - The quadratic equation has real roots because $\Delta \geq 0$ under the given conditions. - The conditions restrict $a$ and $b$ to a region where the quadratic behaves predictably. 12. **Additional note:** The relation $AB^2 = AE \cdot AD$ and the description of the graph as a circle-like emblem suggest a geometric interpretation, possibly involving lengths in a circle or ellipse, but more context is needed for detailed analysis. **Final answer:** The quadratic equation has real roots for $x$ under the given conditions on $a$ and $b$ because the discriminant $$\Delta = -8(a^2 + ab + b^2) + 12 \geq 0.$$