We are given the quadratic equation:

$k^2 + 22k - 21 = 0$

To solve by completing the square, follow these steps:

Step 1: Move the constant to the other side

First, move the constant term (-21) to the other side:

$k^2 + 22k = 21$

Step 2: Complete the square

Next, to complete the square, take half of the coefficient of $k$ (which is 22), square it, and then add it to both sides of the equation.

Half of 22 is $\frac{22}{2} = 11$, and squaring it gives:

$11^2 = 121$

Add this value to both sides:

$k^2 + 22k + 121 = 21 + 121$

This simplifies to:

$(k + 11)^2 = 142$

Step 3: Solve for $k$

Now, take the square root of both sides:

$k + 11 = \pm \sqrt{142}$

Solve for $k$:

$k = -11 \pm \sqrt{142}$

Step 4: Approximate the values

To find the approximate values of $k$, calculate $\sqrt{142}$:

$\sqrt{142} \approx 11.91$

Thus, the solutions are:

$k = -11 + 11.91 \approx 0.91$ and $k = -11 - 11.91 \approx -22.91$

Final Answer:

The solutions are approximately:

$k \approx 0.91 \quad \text{or} \quad k \approx -22.91$

Would you like further explanation or details? Here are some related questions for you:

1. How can completing the square be applied to other types of quadratic equations?
2. What is the relationship between completing the square and the quadratic formula?
3. What happens if the coefficient of $k^2$ is not 1?
4. How do you solve a quadratic equation by factoring?
5. Can completing the square be used for equations that are not quadratic?

Tip: When completing the square, remember to always add the square of half the coefficient of $x$ to both sides of the equation to maintain equality.