Question

Play the game! Find two numbers that multiply to make the 1st number and add to make the 2nd number. Multiply -36; Add -16

Answer 1

Let's find two numbers \( x \) and \( y \) such that:

1. \( x \cdot y = -36 \)
2. \( x + y = -16 \)

To find these numbers, we can express one variable in terms of the other using the second equation:

\[
y = -16 - x
\]

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

\[
x \cdot (-16 - x) = -36
\]

Expanding this equation gives us:

\[
-16x - x^2 = -36
\]

Rearranging it leads to:

\[
x^2 + 16x - 36 = 0
\]

Next, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 16 \), and \( c = -36 \):

Calculating the discriminant:

\[
b^2 - 4ac = 16^2 - 4 \cdot 1 \cdot (-36) = 256 + 144 = 400
\]

Now, applying the quadratic formula:

\[
x = \frac{-16 \pm \sqrt{400}}{2 \cdot 1} = \frac{-16 \pm 20}{2}
\]

Calculating the two possible values for \( x \):

1. \( x = \frac{-16 + 20}{2} = \frac{4}{2} = 2 \)
2. \( x = \frac{-16 - 20}{2} = \frac{-36}{2} = -18 \)

Thus, the possible values for \( x \) are \( 2 \) and \( -18 \). Now we can find the corresponding values for \( y \).

1. If \( x = 2 \):
\[
y = -16 - 2 = -18
\]

2. If \( x = -18 \):
\[
y = -16 - (-18) = 2
\]

So the two numbers are \( 2 \) and \( -18 \).

Verifying:

- Multiply: \( 2 \cdot (-18) = -36 \) (correct)
- Add: \( 2 + (-18) = -16 \) (correct)

Thus, the two numbers you are looking for are \( 2 \) and \( -18 \).