To solve the quadratic equation \( x^2 + 9x - 22 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, the coefficients are:
- \( a = 1 \)
- \( b = 9 \)
- \( c = -22 \)
Now, we can calculate the discriminant (\( b^2 - 4ac \)):
\[ b^2 - 4ac = 9^2 - 4(1)(-22) = 81 + 88 = 169 \]
Next, we can substitute \( a \), \( b \), and the discriminant back into the quadratic formula:
\[ x = \frac{-9 \pm \sqrt{169}}{2 \cdot 1} = \frac{-9 \pm 13}{2} \]
Now we compute the two potential solutions:
-
Using the positive sign: \[ x = \frac{-9 + 13}{2} = \frac{4}{2} = 2 \]
-
Using the negative sign: \[ x = \frac{-9 - 13}{2} = \frac{-22}{2} = -11 \]
The positive solution is:
\[ \boxed{2} \]