To solve the quadratic equation \(x^2 - 6x - 16 = 0\) using the quadratic formula, we use the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, \(a = 1\), \(b = -6\), and \(c = -16\).
First, we calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = (-6)^2 - 4(1)(-16) = 36 + 64 = 100 \]
Now, we can plug in the values into the quadratic formula:
\[ x = \frac{-(-6) \pm \sqrt{100}}{2(1)} = \frac{6 \pm 10}{2} \]
This gives us two solutions:
-
Using the positive square root: \[ x = \frac{6 + 10}{2} = \frac{16}{2} = 8 \]
-
Using the negative square root: \[ x = \frac{6 - 10}{2} = \frac{-4}{2} = -2 \]
Thus, the solutions to the equation \(x^2 - 6x - 16 = 0\) are:
\[ x = 8 \quad \text{and} \quad x = -2 \]
Therefore, the correct answer is:
D. \(x=8, x=-2\)