Find two numbers that have a sum of 23 and a product of 120

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.

Find two numbers that have a sum of 23 and a product of 120.

Explore how to find two numbers that add up to 23 and multiply to 120 using quadratic equations. This step-by-step guide breaks down key algebraic concepts, including the quadratic formula and factoring techniques. Perfect for students in Grades 7-9.

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