To solve the quadratic equation \( 2x^2 + x - 6 = 0 \) by completing the square, follow these steps:

### Step 1: Divide the equation by the coefficient of \( x^2 \)
First, divide every term by 2 (the coefficient of \( x^2 \)) to simplify the equation:
\[
x^2 + \frac{1}{2}x - 3 = 0
\]

### Step 2: Move the constant term to the other side of the equation
Add 3 to both sides:
\[
x^2 + \frac{1}{2}x = 3
\]

### Step 3: Complete the square
To complete the square, we need to add and subtract the square of half the coefficient of \( x \) to the left side. The coefficient of \( x \) is \( \frac{1}{2} \), so half of this is \( \frac{1}{4} \), and squaring this gives \( \left(\frac{1}{4}\right)^2 = \frac{1}{16} \).

Add \( \frac{1}{16} \) to both sides of the equation:
\[
x^2 + \frac{1}{2}x + \frac{1}{16} = 3 + \frac{1}{16}
\]

### Step 4: Write the left side as a perfect square
Now, the left side is a perfect square:
\[
\left(x + \frac{1}{4}\right)^2 = \frac{48}{16} + \frac{1}{16} = \frac{49}{16}
\]

### Step 5: Solve for \( x \)
Take the square root of both sides:
\[
x + \frac{1}{4} = \pm \frac{7}{4}
\]

Subtract \( \frac{1}{4} \) from both sides to solve for \( x \):
\[
x = -\frac{1}{4} \pm \frac{7}{4}
\]

### Step 6: Simplify the solutions
The two possible solutions are:
\[
x = \frac{6}{4} = \frac{3}{2} \quad \text{or} \quad x = -\frac{8}{4} = -2
\]

So, the solutions to the equation are:
\[
x = \frac{3}{2} \quad \text{or} \quad x = -2
\]

Would you like more details on any step, or do you have any questions?

### Related Questions:
1. How do you solve a quadratic equation using the quadratic formula?
2. What are the advantages of completing the square versus other methods?
3. How do you find the vertex of a quadratic function by completing the square?
4. What is the geometric interpretation of completing the square?
5. Can you complete the square for an equation with complex coefficients?

### Tip:
Completing the square is particularly useful for deriving the quadratic formula and understanding the properties of quadratic functions, such as finding the vertex and axis of symmetry.

Learn how to solve the quadratic equation 2x² + x - 6 by completing the square with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic equations and completing the square method. Suitable for students in Grades 10-12.

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