Question

Play the game! Find two numbers that multiply to make the 1st number and add to make the 2nd number. Multiply -48; Add -13

Answer 1

To find two numbers that multiply to -48 and add to -13, we can denote the two numbers as \( x \) and \( y \). We need to solve the following equations:

1. \( x \cdot y = -48 \)
2. \( x + y = -13 \)

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

\[
y = -13 - x
\]

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

\[
x(-13 - x) = -48
\]

Expanding this gives us:

\[
-x^2 - 13x = -48
\]

Rearranging to form a standard quadratic equation:

\[
x^2 + 13x - 48 = 0
\]

Next, we can use the quadratic formula to find the roots. The quadratic formula is given by:

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

Here, \( a = 1 \), \( b = 13 \), and \( c = -48 \).

Calculating the discriminant:

\[
b^2 - 4ac = 13^2 - 4(1)(-48) = 169 + 192 = 361
\]

Now substituting into the quadratic formula:

\[
x = \frac{-13 \pm \sqrt{361}}{2 \cdot 1} = \frac{-13 \pm 19}{2}
\]

We have two potential solutions for \( x \):

1. \( x = \frac{6}{2} = 3 \)
2. \( x = \frac{-32}{2} = -16 \)

Now, we calculate the corresponding \( y \) values for each \( x \):

1. If \( x = 3 \):
\[
y = -13 - 3 = -16
\]

2. If \( x = -16 \):
\[
y = -13 - (-16) = 3
\]

Thus, the two numbers are \( 3 \) and \( -16 \).

So the final answer is:
**3 and -16**, which multiply to make -48 and add to make -13.