1. **State the problem:** Solve the quadratic equation in two variables: $5x^2 - 5xy + 3y^2 = 0$. 2. **Formula and approach:** This is a homogeneous quadratic equation. To find solutions, we can treat it as a quadratic in $x$ with parameter $y$, or vice versa. We look for values of $x$ and $y$ (not both zero) satisfying the equation. 3. **Rewrite the equation:** $$5x^2 - 5xy + 3y^2 = 0$$ 4. **Divide both sides by $y^2$ (assuming $y \neq 0$):** $$5\left(\frac{x}{y}\right)^2 - 5\left(\frac{x}{y}\right) + 3 = 0$$ Let $t = \frac{x}{y}$. 5. **Solve the quadratic in $t$:** $$5t^2 - 5t + 3 = 0$$ 6. **Calculate the discriminant:** $$\Delta = (-5)^2 - 4 \times 5 \times 3 = 25 - 60 = -35$$ 7. **Interpretation:** Since $\Delta < 0$, there are no real solutions for $t$, so no real ratio $\frac{x}{y}$ satisfies the equation except the trivial solution $x = y = 0$. 8. **Conclusion:** The only real solution is the trivial one: $x = 0$ and $y = 0$. If complex solutions are allowed, solve using complex numbers: $$t = \frac{5 \pm \sqrt{-35}}{2 \times 5} = \frac{5 \pm i\sqrt{35}}{10} = \frac{1}{2} \pm \frac{i\sqrt{35}}{10}$$ Thus, $$x = t y = \left(\frac{1}{2} \pm \frac{i\sqrt{35}}{10}\right) y$$ **Final answer:** - Real solutions: $x = 0$, $y = 0$ - Complex solutions: $x = \left(\frac{1}{2} \pm \frac{i\sqrt{35}}{10}\right) y$ for any $y \neq 0$.