To solve the quadratic equation \( x^2 - 6x - 16 = 0 \) using the quadratic formula, we first identify the coefficients from 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 we will substitute the values of \( a \), \( b \), and \( c \) into the formula.
- Calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-6)^2 - 4(1)(-16) = 36 + 64 = 100 \]
- Now substitute \( b \) and the discriminant into the quadratic formula:
\[ x = \frac{-(-6) \pm \sqrt{100}}{2(1)} = \frac{6 \pm 10}{2} \]
- This gives us two possible 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:
\[ x = 8 \quad \text{and} \quad x = -2 \]