Write two numbers that multiply to the value on top and add to the value on bottom

Question

8

-6

Answer 1

To find two numbers that multiply to 8 and add to -6, we can set up the equations:

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

1. \( x \cdot y = 8 \)
2. \( x + y = -6 \)

From the second equation, we can express \( y \) in terms of \( x \): \[ y = -6 - x \]

Now, substitute \( y \) in the first equation: \[ x(-6 - x) = 8 \] \[ -6x - x^2 = 8 \] Rearranging gives: \[ x^2 + 6x + 8 = 0 \]

Now we can factor the quadratic equation: To factor \( x^2 + 6x + 8 \), we look for two numbers that multiply to 8 and add to 6. The numbers are 2 and 4.

Thus, we factor it as: \[ (x + 2)(x + 4) = 0 \]

So, set each factor to zero:

1. \( x + 2 = 0 \) → \( x = -2 \)
2. \( x + 4 = 0 \) → \( x = -4 \)

Thus, the two numbers are \( -2 \) and \( -4 \).

Checking:

• Multiply: \( -2 \times -4 = 8 \)
• Add: \( -2 + -4 = -6 \)

So the two numbers are \( -2 \) and \( -4 \).