https://file.aiquickdraw.com/mathbot/file/1725790629693k4uun3m2.jpg

Let's solve the quadratic equations provided in the image using the method of completing the square:

### 1. \( x^2 + 6x + 5 = 0 \)

**Step 1: Move the constant term to the other side.**
\[
x^2 + 6x = -5
\]

**Step 2: Complete the square.**
To complete the square, we take half the coefficient of \(x\), which is \(6\), divide it by 2 to get \(3\), and then square it, giving us \(9\).

Add \(9\) to both sides of the equation:
\[
x^2 + 6x + 9 = -5 + 9
\]
\[
(x + 3)^2 = 4
\]

**Step 3: Solve for \(x\).**
Take the square root of both sides:
\[
x + 3 = \pm 2
\]

Now solve for \(x\):
\[
x = -3 \pm 2
\]

So the two solutions are:
\[
x = -3 + 2 = -1 \quad \text{or} \quad x = -3 - 2 = -5
\]

Thus, the solutions are \(x = -1\) and \(x = -5\).

---

### 2. \( x^2 + 12x + 36 = 0 \)

**Step 1: Move the constant term to the other side.**
\[
x^2 + 12x = -36
\]

**Step 2: Complete the square.**
Take half of \(12\), divide it by 2 to get \(6\), and square it, giving \(36\).

Add \(36\) to both sides:
\[
x^2 + 12x + 36 = -36 + 36
\]
\[
(x + 6)^2 = 0
\]

**Step 3: Solve for \(x\).**
Take the square root of both sides:
\[
x + 6 = 0
\]

Thus, the solution is:
\[
x = -6
\]

---

### Final Answers:
1. \(x = -1\) and \(x = -5\)
2. \(x = -6\)

Would you like further clarification on any of the steps or more details?

### Related Questions:
1. How does completing the square differ from using the quadratic formula?
2. Can all quadratic equations be solved by completing the square?
3. Why is it necessary to add the square term on both sides of the equation?
4. What are the advantages of completing the square in solving quadratic equations?
5. How can we use completing the square to derive the quadratic formula?

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

Find the solutions of each of the following quadratic equations by completing the square: 1. x^2 + 6x + 5 = 0, 2. x^2 + 12x + 36 = 0

Learn how to solve quadratic equations by completing the square. This example covers x^2 + 6x + 5 = 0 and x^2 + 12x + 36 = 0 with step-by-step solutions. Master key algebra concepts suitable for Grades 8-10.

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