To find the solutions to the quadratic equation \(x^2 - 6x - 16 = 0\) using the quadratic formula, we can start by identifying the coefficients in the standard form \(ax^2 + bx + c = 0\):
- \(a = 1\)
- \(b = -6\)
- \(c = -16\)
The quadratic formula is given by:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
Now, let's plug in the values of \(a\), \(b\), and \(c\):
- Calculate \(b^2 - 4ac\):
\[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100 \]
- Now apply the quadratic formula:
\[ x = \frac{{-(-6) \pm \sqrt{100}}}{2 \cdot 1} \]
Simplifying this gives:
\[ x = \frac{{6 \pm 10}}{2} \]
Now we have two potential solutions:
- \(x = \frac{{6 + 10}}{2} = \frac{16}{2} = 8\)
- \(x = \frac{{6 - 10}}{2} = \frac{-4}{2} = -2\)
Thus, the solutions to the equation \(x^2 - 6x - 16 = 0\) are:
\[ x = 8 \quad \text{and} \quad x = -2 \]