Question

Write two numbers that multiply to the value on top and add to the value on bottom

-16

-6

Answer 1

To find two numbers that multiply to -16 and add to -6, let's denote the numbers as \( x \) and \( y \). We need to solve the following system of equations:

1. \( xy = -16 \) (the product)
2. \( x + y = -6 \) (the sum)

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

\[
y = -6 - x
\]

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

\[
x(-6 - x) = -16
\]

Expanding this, we get:

\[
-x^2 - 6x = -16
\]

Rearranging gives:

\[
x^2 + 6x - 16 = 0
\]

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

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

where \( a = 1 \), \( b = 6 \), and \( c = -16 \):

\[
x = \frac{-6 \pm \sqrt{6^2 - 4 \times 1 \times (-16)}}{2 \times 1}
\]

Calculating the discriminant:

\[
x = \frac{-6 \pm \sqrt{36 + 64}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{100}}{2}
\]
\[
x = \frac{-6 \pm 10}{2}
\]

This gives us two potential solutions:

1. \( x = \frac{4}{2} = 2 \)
2. \( x = \frac{-16}{2} = -8 \)

Thus, \( x = 2 \) and \( y = -8 \) or vice versa.

The two numbers that multiply to -16 and add to -6 are \( 2 \) and \( -8 \).