Let's solve the quadratic equation using the method of completing the square.

The equation provided is: $x^2 + 10x + 22 = 0$

Step 1: Move the constant term to the right side

First, subtract 22 from both sides to move the constant term to the right-hand side: $x^2 + 10x = -22$

Step 2: Complete the square

Now, we need to complete the square on the left side. To do this, take half of the coefficient of $x$ (which is 10), square it, and add it to both sides.

$\left(\frac{10}{2}\right)^2 = 25$

Add 25 to both sides: $x^2 + 10x + 25 = -22 + 25$ $(x + 5)^2 = 3$

Step 3: Solve for $x$

Now, take the square root of both sides: $x + 5 = \pm \sqrt{3}$

Finally, solve for $x$: $x = -5 \pm \sqrt{3}$

Thus, the solutions are: $x = -5 + \sqrt{3}, \quad x = -5 - \sqrt{3}$

Answers for the form and solution boxes:

• Form: $(x + 5)^2 = 3$
• Solution: $x = -5 + \sqrt{3}, -5 - \sqrt{3}$

Would you like any further details or explanations?

Here are 5 related questions to expand your understanding:

1. What is the geometric meaning of completing the square?
2. How does completing the square relate to the quadratic formula?
3. Can all quadratic equations be solved using the method of completing the square?
4. How do you simplify expressions involving square roots, like $\sqrt{3}$?
5. What is the significance of the plus-minus sign ($\pm$) when solving quadratic equations?

Tip: When completing the square, always remember to balance the equation by adding the same value to both sides.