To solve the quadratic equation $2x^2 + 3x - 5 = 0$ by completing the square, follow these steps:

Step 1: Move the constant term to the other side

First, we rewrite the equation as: $2x^2 + 3x = 5$

Step 2: Divide the equation by the coefficient of $x^2$

To complete the square, we need the coefficient of $x^2$ to be 1, so divide the entire equation by 2: $x^2 + \frac{3}{2}x = \frac{5}{2}$

Step 3: Complete the square

Now, take half of the coefficient of $x$, which is $\frac{3}{2}$, and square it. $\left(\frac{3}{4}\right)^2 = \frac{9}{16}$ Add $\frac{9}{16}$ to both sides of the equation: $x^2 + \frac{3}{2}x + \frac{9}{16} = \frac{5}{2} + \frac{9}{16}$

Step 4: Simplify the right-hand side

To simplify the right-hand side, express $\frac{5}{2}$ as $\frac{40}{16}$, then add: $\frac{40}{16} + \frac{9}{16} = \frac{49}{16}$ So, the equation becomes: $\left(x + \frac{3}{4}\right)^2 = \frac{49}{16}$

Step 5: Solve for $x$

Take the square root of both sides: $x + \frac{3}{4} = \pm \frac{7}{4}$

Step 6: Solve for $x$

Now, solve for $x$ by subtracting $\frac{3}{4}$ from both sides: $x = -\frac{3}{4} \pm \frac{7}{4}$ This gives two solutions: $x = \frac{-3 + 7}{4} = 1 \quad \text{or} \quad x = \frac{-3 - 7}{4} = -\frac{5}{2}$

Final Solution:

The solutions are: $x = 1 \quad \text{or} \quad x = -\frac{5}{2}$

Do you want more details or have any questions about this method?

Here are 5 questions that expand on this topic:

1. What is the difference between completing the square and factoring?
2. Can completing the square be used for any quadratic equation?
3. How can the method of completing the square be used to derive the quadratic formula?
4. What is the geometric interpretation of completing the square?
5. How does completing the square help in solving equations with complex numbers?

Tip: Completing the square is particularly useful for converting a quadratic equation into vertex form, which can be helpful for graphing parabolas.