To solve the quadratic equation \(x^2 - x - 30 = 0\) by completing the square, follow these steps:

### Step 1: Move the constant term to the right side of the equation
Start by isolating the quadratic and linear terms on the left side:

\[
x^2 - x = 30
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \(x\), square it, and then add it to both sides of the equation.

The coefficient of \(x\) is \(-1\). Half of \(-1\) is \(-\frac{1}{2}\), and squaring it gives:

\[
\left(-\frac{1}{2}\right)^2 = \frac{1}{4}
\]

Add \(\frac{1}{4}\) to both sides of the equation:

\[
x^2 - x + \frac{1}{4} = 30 + \frac{1}{4}
\]

### Step 3: Rewrite the left side as a perfect square trinomial
The left side of the equation is now a perfect square trinomial:

\[
\left(x - \frac{1}{2}\right)^2 = 30 + \frac{1}{4}
\]

Simplify the right side:

\[
\left(x - \frac{1}{2}\right)^2 = \frac{120}{4} + \frac{1}{4} = \frac{121}{4}
\]

### Step 4: Take the square root of both sides
To solve for \(x\), take the square root of both sides:

\[
x - \frac{1}{2} = \pm \sqrt{\frac{121}{4}}
\]

\[
x - \frac{1}{2} = \pm \frac{11}{2}
\]

### Step 5: Solve for \(x\)
Now, solve for \(x\) by adding \(\frac{1}{2}\) to both sides:

1. \(x = \frac{1}{2} + \frac{11}{2} = \frac{12}{2} = 6\)
2. \(x = \frac{1}{2} - \frac{11}{2} = \frac{-10}{2} = -5\)

### Final Solutions
The solutions are:
\[
x = 6 \quad \text{or} \quad x = -5
\]

Would you like more details or have any questions?

Here are five related questions you might find interesting:
1. How would you solve \(x^2 + 6x + 9 = 0\) by completing the square?
2. What are the advantages of completing the square over other methods for solving quadratic equations?
3. Can you use completing the square to find the vertex of the parabola \(y = x^2 - x - 30\)?
4. How does the method of completing the square lead to the quadratic formula?
5. How can you apply completing the square to solve the equation \(2x^2 + 3x - 4 = 0\)?

**Tip:** Completing the square is also useful for rewriting a quadratic equation in vertex form, which helps in graphing the function and identifying its vertex directly.

Learn how to solve the quadratic equation x^2 - x - 30 = 0 using the method of completing the square. This detailed step-by-step solution covers completing the square, quadratic formula application, and solving quadratic equations. Suitable for high school students (Grades 9-12).

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