To find two numbers that multiply to -16 and add to -6, let's denote the numbers as \( x \) and \( y \). We need to solve the following system of equations:
- \( xy = -16 \) (the product)
- \( x + y = -6 \) (the sum)
From the second equation, we can express \( y \) in terms of \( x \):
\[ y = -6 - x \]
Now, we can substitute this expression for \( y \) into the first equation:
\[ x(-6 - x) = -16 \]
Expanding this, we get:
\[ -x^2 - 6x = -16 \]
Rearranging gives:
\[ x^2 + 6x - 16 = 0 \]
Next, we can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = 6 \), and \( c = -16 \):
\[ x = \frac{-6 \pm \sqrt{6^2 - 4 \times 1 \times (-16)}}{2 \times 1} \]
Calculating the discriminant:
\[ x = \frac{-6 \pm \sqrt{36 + 64}}{2} \] \[ x = \frac{-6 \pm \sqrt{100}}{2} \] \[ x = \frac{-6 \pm 10}{2} \]
This gives us two potential solutions:
- \( x = \frac{4}{2} = 2 \)
- \( x = \frac{-16}{2} = -8 \)
Thus, \( x = 2 \) and \( y = -8 \) or vice versa.
The two numbers that multiply to -16 and add to -6 are \( 2 \) and \( -8 \).
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