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The problem states:

"Two numbers multiply together to give \(-20\) and add together to give \(-1\). What are the two numbers?"

Let's solve this step by step.

We need to find two numbers, say \( x \) and \( y \), that satisfy:

1. \( x \times y = -20 \)
2. \( x + y = -1 \)

To find these numbers, we can use the following approach:

### Step 1: Express one variable in terms of the other
From the second equation, we can express \( y \) in terms of \( x \):
\[
y = -1 - x
\]

### Step 2: Substitute into the first equation
Now, substitute this expression for \( y \) into the first equation:
\[
x \times (-1 - x) = -20
\]
Simplify:
\[
-x - x^2 = -20
\]
Rearrange this to get a quadratic equation:
\[
x^2 + x - 20 = 0
\]

### Step 3: Solve the quadratic equation
Now, we solve the quadratic equation \( x^2 + x - 20 = 0 \). We can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For this equation, \( a = 1 \), \( b = 1 \), and \( c = -20 \).

Substitute these values into the quadratic formula:
\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-20)}}{2(1)}
\]
\[
x = \frac{-1 \pm \sqrt{1 + 80}}{2}
\]
\[
x = \frac{-1 \pm \sqrt{81}}{2}
\]
\[
x = \frac{-1 \pm 9}{2}
\]

### Step 4: Calculate the two solutions
The two possible values for \( x \) are:
1. \( x = \frac{-1 + 9}{2} = \frac{8}{2} = 4 \)
2. \( x = \frac{-1 - 9}{2} = \frac{-10}{2} = -5 \)

### Step 5: Find the corresponding values for \( y \)
For \( x = 4 \), \( y = -1 - 4 = -5 \).
For \( x = -5 \), \( y = -1 - (-5) = 4 \).

Thus, the two numbers are \( 4 \) and \( -5 \).

Would you like more details or have any other questions?

Here are 5 related questions to expand on this topic:
1. How would you solve a quadratic equation by factoring?
2. Can you think of another pair of numbers that multiply to a different negative product but add to \(-1\)?
3. How do you use the discriminant to determine the number of real solutions for a quadratic equation?
4. What are the steps to complete the square for solving quadratic equations?
5. How can you apply the quadratic formula to solve more complex equations?

**Tip:** Always check your solutions by substituting them back into the original equations!

Two numbers multiply together to give -20 and add together to give -1. What are the two numbers?

Learn how to find two numbers that multiply to -20 and add up to -1. This problem involves key algebraic concepts, including quadratic equations and the quadratic formula. Suitable for students in Grades 8-10.

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