Solve the equation x2 + 11x − 24 = 0 by completing the square.

The coefficient of $x$ is 11.
Half of 11 is $\frac{11}{2}$.
Squaring it gives $\left(\frac{11}{2}\right)^2 = \frac{121}{4}$.
Add $\frac{121}{4}$ to both sides:

$$x^2 + 11x + \frac{121}{4} = 24 + \frac{121}{4}$$

The left side factors as $\left(x + \frac{11}{2}\right)^2$.
For the right side, find a common denominator: $24 = \frac{24 \times 4}{4} = \frac{96}{4}$.
So, $24 + \frac{121}{4} = \frac{96}{4} + \frac{121}{4} = \frac{96 + 121}{4} = \frac{217}{4}$.
The equation becomes:

$$\left(x + \frac{11}{2}\right)^2 = \frac{217}{4}$$

$$x + \frac{11}{2} = \pm\sqrt{\frac{217}{4}}$$ $$x + \frac{11}{2} = \pm\frac{\sqrt{217}}{\sqrt{4}}$$ $$x + \frac{11}{2} = \pm\frac{\sqrt{217}}{2}$$

$$x = -\frac{11}{2} \pm \frac{\sqrt{217}}{2}$$ $$x = \frac{-11 \pm \sqrt{217}}{2}$$

The solutions are:

$$x_1 = \frac{-11 + \sqrt{217}}{2}$$ $$x_2 = \frac{-11 - \sqrt{217}}{2}$$