Given the problem, we need to find two numbers $x$ and $y$ such that:

1. Their sum is 11: $x + y = 11$
2. Their product is 30: $xy = 30$

We can solve this problem using a system of equations.

First, express $y$ in terms of $x$ using the first equation: $y = 11 - x$

Next, substitute $y$ into the second equation: $x(11 - x) = 30$ $11x - x^2 = 30$ $x^2 - 11x + 30 = 0$

This is a quadratic equation. We can solve for $x$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -11$, and $c = 30$.

Substitute the values into the formula: $x = \frac{11 \pm \sqrt{(-11)^2 - 4 \cdot 1 \cdot 30}}{2 \cdot 1}$ $x = \frac{11 \pm \sqrt{121 - 120}}{2}$ $x = \frac{11 \pm \sqrt{1}}{2}$ $x = \frac{11 \pm 1}{2}$

So, we have two solutions for $x$: $x = \frac{12}{2} = 6$ $x = \frac{10}{2} = 5$

Therefore, the two numbers are $x = 6$ and $y = 5$, or $x = 5$ and $y = 6$.

The numbers are $\boxed{5 \text{ and } 6}$.

Would you like details on this solution or have any questions?

Here are 8 related questions that expand on this information:

1. What are the properties of quadratic equations?
2. How do you derive the quadratic formula?
3. Can a quadratic equation have complex roots? If so, under what conditions?
4. What are the different methods to solve a system of equations?
5. How can you determine if a given quadratic equation can be factored easily?
6. How do you solve a quadratic equation by completing the square?
7. What is the significance of the discriminant in a quadratic equation?
8. How can you verify the solutions of a quadratic equation?

Tip: When solving quadratic equations, always check if the equation can be factored easily before using the quadratic formula. This can save time and simplify the process.