We are given two pieces of information about two numbers:

1. Their product is 30.
2. Their sum is 11.

Let the two numbers be $x$ and $y$. The information gives us the following system of equations:

$x \cdot y = 30 \quad \text{(1)}$ $x + y = 11 \quad \text{(2)}$

To solve for $x$ and $y$, we can use substitution or the quadratic formula. Let's solve using substitution:

From equation (2), solve for $y$:

$y = 11 - x$

Substitute this into equation (1):

$x \cdot (11 - x) = 30$

Expand the equation:

$x(11) - x^2 = 30$ $11x - x^2 = 30$

Rearrange the equation:

$x^2 - 11x + 30 = 0$

This is a quadratic equation. We can solve it using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $x^2 - 11x + 30 = 0$, the coefficients are $a = 1$, $b = -11$, and $c = 30$. Substituting into the quadratic formula:

$x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(1)(30)}}{2(1)}$ $x = \frac{11 \pm \sqrt{121 - 120}}{2}$ $x = \frac{11 \pm \sqrt{1}}{2}$ $x = \frac{11 \pm 1}{2}$

So, the two possible values for $x$ are:

$x = \frac{11 + 1}{2} = 6 \quad \text{or} \quad x = \frac{11 - 1}{2} = 5$

If $x = 6$, then from equation (2), $y = 11 - 6 = 5$.

If $x = 5$, then $y = 11 - 5 = 6$.

Thus, the two numbers are 5 and 6.

Would you like a further explanation or details on solving quadratic equations?

Here are some related questions to consider:

1. How do we derive the quadratic equation from a system of equations?
2. What happens if the product and sum of the numbers were negative?
3. Can we solve for more than two variables in a similar system of equations?
4. How do you check the solutions to a quadratic equation?
5. What is the significance of the discriminant in the quadratic formula?

Tip: When solving a quadratic equation, always check if the discriminant (the expression under the square root) is positive, zero, or negative. It helps you determine the number and type of solutions!