Question

The sum of two numbers is 22, and the product of the same two numbers is 120. What are the two numbers? Input the lesser number first. (1 point)
and

Answer 1

Let's denote the two numbers as \( x \) and \( y \). According to the problem, we have the following equations:

1. \( x + y = 22 \) (the sum of the numbers)
2. \( xy = 120 \) (the product of the numbers)

We can express \( y \) in terms of \( x \) using the first equation:

\[
y = 22 - x
\]

Now, we can substitute this expression for \( y \) into the second equation:

\[
x(22 - x) = 120
\]

Expanding this, we get:

\[
22x - x^2 = 120
\]

Rearranging gives us a standard form quadratic equation:

\[
x^2 - 22x + 120 = 0
\]

To solve for \( x \), we can use the quadratic formula:

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

In our equation, \( a = 1 \), \( b = -22 \), and \( c = 120 \). Substituting these values in:

\[
x = \frac{22 \pm \sqrt{(-22)^2 - 4 \cdot 1 \cdot 120}}{2 \cdot 1}
\]
\[
x = \frac{22 \pm \sqrt{484 - 480}}{2}
\]
\[
x = \frac{22 \pm \sqrt{4}}{2}
\]
\[
x = \frac{22 \pm 2}{2}
\]

This gives us two possible solutions for \( x \):

1. \( x = \frac{24}{2} = 12 \)
2. \( x = \frac{20}{2} = 10 \)

Thus, the two numbers \( x \) and \( y \) are \( 10 \) and \( 12 \). Since we want to input the lesser number first, the answer is:

10 and 12.