To solve the quadratic equation $3x^2 - 9x - 2 = 0$ by completing the square, follow these steps:

Step 1: Move the constant term to the other side

First, rewrite the equation by moving the constant term $-2$ to the other side:

$3x^2 - 9x = 2$

Step 2: Factor out the coefficient of $x^2$ from the left side

Factor out the coefficient of $x^2$, which is $3$:

$3(x^2 - 3x) = 2$

Step 3: Complete the square

To complete the square inside the parentheses, take half of the coefficient of $x$ (which is $-3$), square it, and add it inside the parentheses:

• Half of $-3$ is $-\frac{3}{2}$.
• Squaring this gives $\left(-\frac{3}{2}\right)^2 = \frac{9}{4}$.

Add and subtract this square inside the parentheses:

$3\left(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\right) = 2$

Simplify inside the parentheses:

$3\left(\left(x - \frac{3}{2}\right)^2 - \frac{9}{4}\right) = 2$

Step 4: Move the subtracted square term to the other side

Distribute the $3$ and then move the $-\frac{27}{4}$ to the right side:

$3\left(x - \frac{3}{2}\right)^2 - \frac{27}{4} = 2$

$3\left(x - \frac{3}{2}\right)^2 = 2 + \frac{27}{4}$

$3\left(x - \frac{3}{2}\right)^2 = \frac{8}{4} + \frac{27}{4} = \frac{35}{4}$

Step 5: Solve for $x$

Divide both sides by 3:

$\left(x - \frac{3}{2}\right)^2 = \frac{35}{12}$

Take the square root of both sides:

$x - \frac{3}{2} = \pm \sqrt{\frac{35}{12}}$

Simplify the square root:

$x - \frac{3}{2} = \pm \frac{\sqrt{35}}{2\sqrt{3}}$

Rationalize the denominator:

$x - \frac{3}{2} = \pm \frac{\sqrt{105}}{6}$

Finally, solve for $x$:

$x = \frac{3}{2} \pm \frac{\sqrt{105}}{6}$

Simplify:

$x = \frac{3 \cdot 3 \pm \sqrt{105}}{6}$

So, the two solutions are:

$x_1 = \frac{9 + \sqrt{105}}{6}, \quad x_2 = \frac{9 - \sqrt{105}}{6}$

These are the same as the solutions obtained using the quadratic formula.

Would you like further details or have any questions?

Here are some related questions:

1. Why is completing the square useful in solving quadratic equations?
2. How can you use completing the square to derive the quadratic formula?
3. What is the geometric interpretation of completing the square?
4. How does completing the square help in graphing parabolas?
5. Can completing the square be used for any quadratic equation?

Tip: Completing the square is particularly useful when you want to find the vertex form of a quadratic equation or when solving integrals involving quadratic expressions.