$$  $$What multiplies to $$ -\frac{19}{8} $$ and adds to $$ -\frac12 $$

To find two numbers that multiply to $-\frac{19}{8}$ and add to $-\frac{1}{2}$, let's denote these two numbers as $ a $ and $ b $. We need to satisfy the following two equations:

\[
a \cdot b = -\frac{19}{8}
\]
\[
a + b = -\frac{1}{2}
\]

### Step 1: Express $ a $ and $ b $ in terms of their sum and product

From the sum equation $ a + b = -\frac{1}{2} $, we can express one of the variables in terms of the other. Let's express $ b $ in terms of $ a $:

\[
b = -\frac{1}{2} - a
\]

### Step 2: Substitute into the product equation

Now substitute $ b = -\frac{1}{2} - a $ into the product equation:

\[
a \cdot \left(-\frac{1}{2} - a\right) = -\frac{19}{8}
\]

Expanding this equation:

\[
-a \cdot \frac{1}{2} - a^2 = -\frac{19}{8}
\]

Multiply through by $-1$ to simplify:

\[
\frac{a}{2} + a^2 = \frac{19}{8}
\]

To eliminate the fraction, multiply the entire equation by 8:

\[
4a + 8a^2 = 19
\]

Rearrange this into a standard quadratic form:

\[
8a^2 + 4a - 19 = 0
\]

### Step 3: Solve the quadratic equation

We can now solve this quadratic equation using the quadratic formula:

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

Here, $ a = 8 $, $ b = 4 $, and $ c = -19 $:

\[
a = \frac{-4 \pm \sqrt{4^2 - 4(8)(-19)}}{2(8)}
\]
\[
a = \frac{-4 \pm \sqrt{16 + 608}}{16}
\]
\[
a = \frac{-4 \pm \sqrt{624}}{16}
\]
\[
a = \frac{-4 \pm 4\sqrt{39}}{16}
\]
\[
a = \frac{-1 \pm \sqrt{39}}{4}
\]

So, the two possible values for $ a $ are:

\[
a = \frac{-1 + \sqrt{39}}{4} \quad \text{or} \quad a = \frac{-1 - \sqrt{39}}{4}
\]

### Step 4: Find the corresponding $ b $ values

Since $ b = -\frac{1}{2} - a $, the corresponding values of $ b $ will be:

\[
b = -\frac{1}{2} - \frac{-1 + \sqrt{39}}{4} = \frac{-2 - \sqrt{39}}{4}
\]
\[
b = -\frac{1}{2} - \frac{-1 - \sqrt{39}}{4} = \frac{-2 + \sqrt{39}}{4}
\]

### Conclusion

The two numbers that multiply to $-\frac{19}{8}$ and add to $-\frac{1}{2}$ are:

\[
\left( \frac{-1 + \sqrt{39}}{4}, \frac{-2 - \sqrt{39}}{4} \right) \quad \text{and} \quad \left( \frac{-1 - \sqrt{39}}{4}, \frac{-2 + \sqrt{39}}{4} \right)
\]

Would you like to explore any specific details or need further assistance?

---

Here are 5 related questions:
1. How can the quadratic formula be derived from a general quadratic equation?
2. What is the significance of the discriminant in the quadratic formula?
3. How can you check if the roots of a quadratic equation are real or complex?
4. What are some practical applications of solving quadratic equations in real-world scenarios?
5. How do you simplify expressions involving square roots like $\sqrt{39}$?

**Tip:** When solving quadratic equations, always check if the equation can be factored before applying the quadratic formula. This can save time and simplify the process.

Explore how to find two numbers that multiply to -19/8 and add to -1/2 using algebraic methods. This detailed solution involves solving a quadratic equation and understanding the relationships between the roots and coefficients. Suitable for high school students.

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