Question

what multiplies to 55 and adds to -16

Answer 1

To find two numbers that multiply to 55 and add to -16, we can set up the equations based on the conditions provided:

Let the two numbers be \(x\) and \(y\).

1. \(x \cdot y = 55\)
2. \(x + y = -16\)

From the second equation, we can express one variable in terms of the other:

\[ y = -16 - x \]

Now, we can substitute this into the first equation:

\[ x(-16 - x) = 55 \]

Expanding this, we get:

\[ -16x - x^2 = 55 \]

Rearranging it to set it to zero:

\[ x^2 + 16x + 55 = 0 \]

Next, we can use the quadratic formula to solve for \(x\):

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

In our equation, \(a = 1\), \(b = 16\), and \(c = 55\):

\[ x = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 1 \cdot 55}}{2 \cdot 1} \]
\[ x = \frac{-16 \pm \sqrt{256 - 220}}{2} \]
\[ x = \frac{-16 \pm \sqrt{36}}{2} \]
\[ x = \frac{-16 \pm 6}{2} \]

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

1. \( x = \frac{-10}{2} = -5 \)
2. \( x = \frac{-22}{2} = -11 \)

Thus, the two numbers are \(-5\) and \(-11\).

To verify:
- They multiply: \((-5) \cdot (-11) = 55\)
- They add: \((-5) + (-11) = -16\)

The two numbers that multiply to 55 and add to -16 are \(-5\) and \(-11\).