To solve the quadratic equation \( x^2 - 6x - 16 = 0 \) using the quadratic formula, we start with the standard form of a quadratic equation:
\[ ax^2 + bx + c = 0 \]
In this case, \( a = 1 \), \( b = -6 \), and \( c = -16 \).
The quadratic formula is given by:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
Let's compute the values step-by-step:
- Calculate \( b^2 - 4ac \):
\[ b^2 = (-6)^2 = 36 \] \[ 4ac = 4 \cdot 1 \cdot (-16) = -64 \] \[ b^2 - 4ac = 36 - (-64) = 36 + 64 = 100 \]
- Now substituting back into the quadratic formula:
\[ x = \frac{{-(-6) \pm \sqrt{100}}}{2 \cdot 1} \] \[ x = \frac{{6 \pm 10}}{2} \]
- This gives us two potential solutions:
\[ x_1 = \frac{{6 + 10}}{2} = \frac{16}{2} = 8 \]
\[ x_2 = \frac{{6 - 10}}{2} = \frac{-4}{2} = -2 \]
Thus, the solutions to the equation \( x^2 - 6x - 16 = 0 \) are:
\[ x = 8, \quad x = -2 \]
So, the correct response is:
x equals 8, x equals negative 2.