To solve the quadratic equation $2x = 1 - 3x^2$ using the method of completing the square, follow these steps:

Step 1: Rearrange the equation

First, rearrange the equation so that all terms are on one side, leaving zero on the other side.

$3x^2 + 2x - 1 = 0$

Step 2: Normalize the quadratic term

To complete the square, the coefficient of $x^2$ should be 1. Here, it is 3, so divide the entire equation by 3.

$x^2 + \frac{2}{3}x - \frac{1}{3} = 0$

Step 3: Move the constant term to the other side

Move the constant term to the right side of the equation.

$x^2 + \frac{2}{3}x = \frac{1}{3}$

Step 4: Complete the square

To complete the square, add and subtract the square of half the coefficient of $x$ to the left side.

The coefficient of $x$ is $\frac{2}{3}$. Half of it is $\frac{1}{3}$, and squaring this gives $\frac{1}{9}$.

Add and subtract $\frac{1}{9}$ to the left side.

$x^2 + \frac{2}{3}x + \frac{1}{9} - \frac{1}{9} = \frac{1}{3}$

This simplifies to:

$\left(x + \frac{1}{3}\right)^2 - \frac{1}{9} = \frac{1}{3}$

Step 5: Isolate the square term

Move $-\frac{1}{9}$ to the right side of the equation.

$\left(x + \frac{1}{3}\right)^2 = \frac{1}{3} + \frac{1}{9}$

Step 6: Simplify the right side

To simplify the right side, find a common denominator.

$\frac{1}{3} + \frac{1}{9} = \frac{3}{9} + \frac{1}{9} = \frac{4}{9}$

So, the equation becomes:

$\left(x + \frac{1}{3}\right)^2 = \frac{4}{9}$

Step 7: Solve for $x$

Take the square root of both sides.

$x + \frac{1}{3} = \pm \frac{2}{3}$

Step 8: Isolate $x$

Solve for $x$ by subtracting $\frac{1}{3}$ from both sides.

$x = -\frac{1}{3} \pm \frac{2}{3}$

This gives two solutions:

$x = \frac{1}{3} \quad \text{or} \quad x = -1$

Final Answer:

The solutions to the equation are:

$x = \frac{1}{3} \quad \text{or} \quad x = -1$

Would you like more details or have any questions?

Related Questions:

1. How can completing the square be used to derive the quadratic formula?
2. What is the geometrical interpretation of completing the square?
3. How does completing the square help in finding the vertex of a parabola?
4. Can you solve a quadratic equation using the quadratic formula and compare it with completing the square?
5. How do you complete the square for an equation with complex coefficients?

Tip: Completing the square is a powerful tool not just for solving quadratics, but also for rewriting expressions in a form that reveals properties like the vertex of a parabola.