Question

What two numbers when multiplied gives 5 but when added gives 16

Answer 1

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

1. \( x \cdot y = 5 \) (the product of the two numbers)
2. \( x + y = 16 \) (the sum of the two numbers)

We can solve these equations by expressing \( y \) in terms of \( x \) using the second equation and substituting it into the first equation.

From the second equation:

\[
y = 16 - x
\]

Now, substituting \( y \) in the first equation:

\[
x(16 - x) = 5
\]

Expanding this gives:

\[
16x - x^2 = 5
\]

Rearranging it to form a standard quadratic equation:

\[
x^2 - 16x + 5 = 0
\]

Now, we can use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -16 \), and \( c = 5 \):

\[
x = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}
\]
\[
x = \frac{16 \pm \sqrt{256 - 20}}{2}
\]
\[
x = \frac{16 \pm \sqrt{236}}{2}
\]

Since \( \sqrt{236} = \sqrt{4 \cdot 59} = 2\sqrt{59} \), we can simplify further:

\[
x = \frac{16 \pm 2\sqrt{59}}{2} = 8 \pm \sqrt{59}
\]

So, the two numbers are:

\[
x = 8 + \sqrt{59} \quad \text{and} \quad y = 8 - \sqrt{59}
\]

Therefore, the two numbers that meet the criteria are \( 8 + \sqrt{59} \) and \( 8 - \sqrt{59} \).