Question

The product of two numbers is 40. If the sum of both number is 13 then what is their difference?

Answer 1

Let the two numbers be \( x \) and \( y \). We have the following system of equations based on the information given:

1. \( xy = 40 \) (the product of the two numbers)
2. \( x + y = 13 \) (the sum of the two numbers)

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

\[
y = 13 - x
\]

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

\[
x(13 - x) = 40
\]

Expanding this, we have:

\[
13x - x^2 = 40
\]

Rearranging gives us a standard quadratic equation:

\[
x^2 - 13x + 40 = 0
\]

To solve for \( x \), we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -13, c = 40 \):

\[
x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4 \cdot 1 \cdot 40}}{2 \cdot 1}
\]

Calculating the discriminant:

\[
(-13)^2 - 4 \cdot 1 \cdot 40 = 169 - 160 = 9
\]

Now substituting back into the formula:

\[
x = \frac{13 \pm \sqrt{9}}{2} = \frac{13 \pm 3}{2}
\]

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

1. \( x = \frac{16}{2} = 8 \)
2. \( x = \frac{10}{2} = 5 \)

From the \( x \) values, we can find the corresponding \( y \) values:

- If \( x = 8 \), then \( y = 13 - 8 = 5 \).
- If \( x = 5 \), then \( y = 13 - 5 = 8 \).

In both cases, the two numbers are \( 8 \) and \( 5 \).

Now, we find their difference:

\[
|x - y| = |8 - 5| = 3
\]

Thus, the difference between the two numbers is \( \boxed{3} \).