To solve the quadratic equation \(x^2 - 8x - 2 = 0\) using the quadratic formula, we can identify the coefficients:
- \(a = 1\)
- \(b = -8\)
- \(c = -2\)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Calculate the discriminant (\(b^2 - 4ac\)):
\[ b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot (-2) = 64 + 8 = 72 \]
- Substitute the values into the quadratic formula:
\[ x = \frac{-(-8) \pm \sqrt{72}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{8 \pm \sqrt{72}}{2} \]
- Simplify \(\sqrt{72}\):
\[ \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \]
- Substitute \(\sqrt{72}\) back into the formula:
\[ x = \frac{8 \pm 6\sqrt{2}}{2} \]
- Simplify further:
\[ x = \frac{8}{2} \pm \frac{6\sqrt{2}}{2} = 4 \pm 3\sqrt{2} \]
Thus, the two solutions are:
\[ x = 4 + 3\sqrt{2} \quad \text{and} \quad x = 4 - 3\sqrt{2} \]