To complete the square for the equation \( 2m^2 + 3m - 3 = 0 \), we follow these steps:

### Step 1: Move the Constant Term
First, isolate the terms involving \( m \) on one side by moving the constant term to the other side:
\[
2m^2 + 3m = 3
\]

### Step 2: Factor Out the Coefficient of \( m^2 \)
Next, factor out the coefficient of \( m^2 \) (which is 2) from the left side:
\[
2 \left( m^2 + \frac{3}{2}m \right) = 3
\]

### Step 3: Complete the Square
Now, to complete the square inside the parentheses, take half of the coefficient of \( m \), square it, and add it inside the parentheses. The coefficient of \( m \) is \( \frac{3}{2} \), so half of that is \( \frac{3}{4} \), and squaring \( \frac{3}{4} \) gives \( \frac{9}{16} \).

Add \( \frac{9}{16} \) inside the parentheses. To keep the equation balanced, we must also subtract \( 2 \times \frac{9}{16} = \frac{9}{8} \) on the right side:
\[
2 \left( m^2 + \frac{3}{2}m + \frac{9}{16} \right) = 3 + \frac{9}{8}
\]

### Step 4: Simplify the Right Side
Convert 3 to a fraction with a denominator of 8:
\[
3 = \frac{24}{8}
\]
So, the equation becomes:
\[
2 \left( m + \frac{3}{4} \right)^2 = \frac{24}{8} + \frac{9}{8}
\]
\[
2 \left( m + \frac{3}{4} \right)^2 = \frac{33}{8}
\]

### Step 5: Divide by 2
Now, divide both sides by 2 to isolate the square term:
\[
\left( m + \frac{3}{4} \right)^2 = \frac{33}{16}
\]

### Step 6: Take the Square Root of Both Sides
Take the square root of both sides, remembering to consider both the positive and negative roots:
\[
m + \frac{3}{4} = \pm \sqrt{\frac{33}{16}}
\]
\[
m + \frac{3}{4} = \pm \frac{\sqrt{33}}{4}
\]

### Step 7: Solve for \( m \)
Finally, isolate \( m \) by subtracting \( \frac{3}{4} \) from both sides:
\[
m = -\frac{3}{4} \pm \frac{\sqrt{33}}{4}
\]

### Final Answer
The solutions are:
\[
m = \frac{-3 + \sqrt{33}}{4} \quad \text{and} \quad m = \frac{-3 - \sqrt{33}}{4}
\]

Would you like a detailed explanation of any specific step, or have any questions?

### Related Questions

1. How would you complete the square for \( 3x^2 + 6x - 5 = 0 \)?
2. What is the vertex form of a quadratic function after completing the square?
3. How does completing the square help in solving quadratic equations?
4. How can you use completing the square to derive the quadratic formula?
5. What are real-world applications of completing the square in mathematics?

#### Tip
Completing the square is especially useful for rewriting quadratic expressions in vertex form, making it easier to analyze the graph's properties, such as its vertex and direction.

This guide explains how to solve the quadratic equation 2m^2 + 3m - 3 = 0 by completing the square. Follow a step-by-step approach covering core algebra concepts, such as isolating terms, factoring out coefficients, and solving for the variable. Perfect for high school students in Grades 9-11.

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