what 2 numbers add to make 8 and multiply to make -105

We need to find two numbers that:

1. Add to give 8.
2. Multiply to give -105.

Let the two numbers be \( x \) and \( y \). Then, we have the system of equations:

\[
x + y = 8 \quad \text{(sum equation)}
\]
\[
x \cdot y = -105 \quad \text{(product equation)}
\]

To solve this, substitute \( y = 8 - x \) from the sum equation into the product equation:

\[
x \cdot (8 - x) = -105
\]

Expanding the equation:

\[
8x - x^2 = -105
\]

Rearrange it into a standard quadratic form:

\[
x^2 - 8x - 105 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(-105)}}{2(1)}
\]
\[
x = \frac{8 \pm \sqrt{64 + 420}}{2}
\]
\[
x = \frac{8 \pm \sqrt{484}}{2}
\]
\[
x = \frac{8 \pm 22}{2}
\]

So, we get two possible values for \( x \):

1. \( x = \frac{8 + 22}{2} = 15 \)
2. \( x = \frac{8 - 22}{2} = -7 \)

Thus, the two numbers are \( 15 \) and \( -7 \).

### Verification:
- \( 15 + (-7) = 8 \)
- \( 15 \times (-7) = -105 \)

So, the two numbers are \( 15 \) and \( -7 \).

Do you want more details or have any questions?

### Relative Questions:
1. How do we solve quadratic equations using the quadratic formula?
2. What are other ways to factor quadratic equations?
3. How can you solve systems of equations with more than two unknowns?
4. How does the discriminant of a quadratic equation determine the nature of its roots?
5. How do you handle quadratic equations with complex roots?

**Tip**: Always verify your solution by substituting the values back into the original equations.

What 2 numbers add to make 8 and multiply to make -105?

Find out how to solve the problem of finding two numbers that add up to 8 and multiply to -105. This detailed guide uses algebraic concepts such as systems of equations and the quadratic formula. Suitable for Grades 8-10 students.

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