Sure! Let's solve this step-by-step:
We need to find two numbers that fulfill the following conditions:
1. Their product is 64.
2. Their sum is -20.
Let's denote these two numbers as [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
We start by writing down the equations given by the problem:
1. [tex]\( x \cdot y = 64 \)[/tex]
2. [tex]\( x + y = -20 \)[/tex]
To find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], we can proceed by considering pairs of factors of 64 and checking if their sum is -20.
First, recall the pairs of factors for 64:
- [tex]\(1 \cdot 64\)[/tex]
- [tex]\(2 \cdot 32\)[/tex]
- [tex]\(4 \cdot 16\)[/tex]
- [tex]\(8 \cdot 8\)[/tex]
- We also include the negative pairs:
- [tex]\((-1) \cdot (-64)\)[/tex]
- [tex]\((-2) \cdot (-32)\)[/tex]
- [tex]\((-4) \cdot (-16)\)[/tex]
- [tex]\((-8) \cdot (-8)\)[/tex]
Now, we test these pairs to see if their sums are -20:
1. [tex]\(1 + 64 = 65\)[/tex]
2. [tex]\(2 + 32 = 34\)[/tex]
3. [tex]\(4 + 16 = 20\)[/tex]
4. [tex]\(8 + 8 = 16\)[/tex]
5. [tex]\((-1) + (-64) = -65\)[/tex]
6. [tex]\((-2) + (-32) = -34\)[/tex]
7. [tex]\((-4) + (-16) = -20\)[/tex]
8. [tex]\((-8) + (-8) = -16\)[/tex]
From these calculations, we see that the pair [tex]\((-4)\)[/tex] and [tex]\((-16)\)[/tex] multiply to 64 and add to -20.
Therefore, the two numbers you are looking for are:
[tex]\[ \boxed{(-16, -4)} \][/tex]
