Let the two numbers be $x$ and $y$. We are given two conditions:

1. The sum of the numbers: $x + y = 23$
2. The product of the numbers: $x \cdot y = 120$

To solve this, we can express $y$ in terms of $x$ using the first equation: $y = 23 - x$

Substitute this into the second equation: $x \cdot (23 - x) = 120$

Now, expand and rearrange the equation: $x(23 - x) = 120 \quad \Rightarrow \quad 23x - x^2 = 120$ $-x^2 + 23x - 120 = 0$

Multiply the entire equation by $-1$ to make it a standard quadratic form: $x^2 - 23x + 120 = 0$

Now, solve this quadratic equation using the quadratic formula: $x = \frac{-(-23) \pm \sqrt{(-23)^2 - 4(1)(120)}}{2(1)}$ $x = \frac{23 \pm \sqrt{529 - 480}}{2}$ $x = \frac{23 \pm \sqrt{49}}{2}$ $x = \frac{23 \pm 7}{2}$

So, the two possible solutions for $x$ are: $x = \frac{23 + 7}{2} = 15 \quad \text{or} \quad x = \frac{23 - 7}{2} = 8$

Thus, the two numbers are 15 and 8.

Would you like a further breakdown or have any other questions?

Here are some related questions:

1. What are the properties of quadratic equations?
2. Can you solve a similar problem with different sum and product values?
3. How does the quadratic formula work in solving equations?
4. What happens if the discriminant is negative in a quadratic equation?
5. How do you factor a quadratic equation directly?

Tip: Always check your solution by substituting the numbers back into the original conditions.