The given quadratic equation is:

$3x^2 + 5x - 2 = 0$

We will solve this equation by completing the square.

Step 1: Move the constant term to the other side of the equation.

$3x^2 + 5x = 2$

Step 2: Divide the entire equation by the coefficient of $x^2$, which is 3, to simplify the quadratic term.

$x^2 + \frac{5}{3}x = \frac{2}{3}$

Step 3: Complete the square.

We need to find the value to add to both sides to make the left-hand side a perfect square trinomial. To do this, take half of the coefficient of $x$ (which is $\frac{5}{3}$), square it, and add it to both sides.

$\text{Half of } \frac{5}{3} = \frac{5}{6}, \quad \left(\frac{5}{6}\right)^2 = \frac{25}{36}$

Now, add $\frac{25}{36}$ to both sides:

$x^2 + \frac{5}{3}x + \frac{25}{36} = \frac{2}{3} + \frac{25}{36}$

Step 4: Simplify the right-hand side.

To add $\frac{2}{3}$ and $\frac{25}{36}$, we need a common denominator. The least common denominator between 3 and 36 is 36.

$\frac{2}{3} = \frac{24}{36}$

So the equation becomes:

$x^2 + \frac{5}{3}x + \frac{25}{36} = \frac{24}{36} + \frac{25}{36} = \frac{49}{36}$

Step 5: Write the left-hand side as a square.

The left-hand side is now a perfect square trinomial:

$\left(x + \frac{5}{6}\right)^2 = \frac{49}{36}$

Step 6: Solve for $x$.

Take the square root of both sides:

$x + \frac{5}{6} = \pm \frac{7}{6}$

Step 7: Solve for $x$ by isolating it.

Now solve for $x$ by subtracting $\frac{5}{6}$ from both sides:

$x = -\frac{5}{6} \pm \frac{7}{6}$

Thus, we have two solutions:

1. $x = -\frac{5}{6} + \frac{7}{6} = \frac{2}{6} = \frac{1}{3}$
2. $x = -\frac{5}{6} - \frac{7}{6} = -\frac{12}{6} = -2$

Final Solutions:

$x = \frac{1}{3}, \quad x = -2$

Would you like a further breakdown of any step, or do you have any questions?

Related Questions:

1. Can you use the quadratic formula on the same equation and compare the results?
2. How does completing the square help in deriving the quadratic formula?
3. What are the benefits of solving quadratic equations by completing the square versus factoring?
4. Can every quadratic equation be solved by completing the square, and why?
5. How can completing the square help in understanding the vertex form of a quadratic equation?

Tip:

Always ensure that the coefficient of $x^2$ is 1 before completing the square to simplify the process.