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The equation you provided is:

\[
x^2 - 6x + 34 = 0
\]

Let's solve this equation using the method of **completing the square**. Here are the steps:

### Step 1: Move the constant term to the right-hand side.
\[
x^2 - 6x = -34
\]

### Step 2: Complete the square.
To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides of the equation. The coefficient of \(x\) is \(-6\), so half of that is \(-3\), and squaring it gives \(9\).

\[
x^2 - 6x + 9 = -34 + 9
\]

\[
(x - 3)^2 = -25
\]

### Step 3: Solve for \(x\).
Take the square root of both sides:

\[
x - 3 = \pm \sqrt{-25}
\]

\[
x - 3 = \pm 5i
\]

### Step 4: Isolate \(x\).
\[
x = 3 \pm 5i
\]

So, the solutions are:

\[
x = 3 + 5i \quad \text{or} \quad x = 3 - 5i
\]

These are complex numbers, as indicated by the imaginary unit \(i = \sqrt{-1}\).

Would you like further details or have any questions?

---

Here are five related questions:
1. What is the general process for solving quadratic equations by completing the square?
2. Can this equation be solved using the quadratic formula instead?
3. How do complex numbers arise in quadratic equations?
4. What is the significance of the imaginary unit \(i\)?
5. Can this method be applied to any quadratic equation?

**Tip:** When completing the square, always ensure that the coefficient of \(x^2\) is 1 before proceeding. If it's not, divide the entire equation by that coefficient.

Solve the equation using the method of completing the square:
x^2 - 6x + 34 = 0

Learn how to solve the quadratic equation x^2 - 6x + 34 using the method of completing the square. This step-by-step solution includes key algebraic concepts, such as completing the square and working with complex numbers. Suitable for high school students in Grades 9-12.

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