Question

The sum of the two numbers is 15 and their product is 56.find the numbers

Answer 1

Let the two numbers be \( x \) and \( y \). We can set up the problem with the following equations based on the information provided:

1. \( x + y = 15 \) (the sum of the numbers)
2. \( x \cdot y = 56 \) (the product of the numbers)

From the first equation, we can express \( y \) in terms of \( x \):

\[
y = 15 - x
\]

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

\[
x \cdot (15 - x) = 56
\]

Expanding this gives:

\[
15x - x^2 = 56
\]

Rearranging the equation leads to:

\[
x^2 - 15x + 56 = 0
\]

Next, we can solve the quadratic equation using the quadratic formula:

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

where \( a = 1 \), \( b = -15 \), and \( c = 56 \).

Calculating the discriminant:

\[
b^2 - 4ac = (-15)^2 - 4 \cdot 1 \cdot 56 = 225 - 224 = 1
\]

Now substituting back into the quadratic formula:

\[
x = \frac{15 \pm \sqrt{1}}{2} = \frac{15 \pm 1}{2}
\]

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

1. \( x = \frac{16}{2} = 8 \)
2. \( x = \frac{14}{2} = 7 \)

Using \( x = 8 \):

\[
y = 15 - 8 = 7
\]

Using \( x = 7 \):

\[
y = 15 - 7 = 8
\]

Thus, the two numbers are \( 7 \) and \( 8 \).

In conclusion, the numbers are **7 and 8**.