Yes! Let's solve the quadratic equation using the completing the square method step by step.

Step 1: Divide the equation by the coefficient of

Since the coefficient of is 3, we divide everything by 3 to simplify:

x^2 - \frac{5}{3}x + \frac{2}{3} = 0

x^2 - \frac{5}{3}x = -\frac{2}{3}

Step 2: Complete the square

To complete the square, we take half the coefficient of , square it, and add it to both sides.

The coefficient of is

Half of is

Squaring it:


\left(-\frac{5}{6}\right)^2 = \frac{25}{36}

Now, add to both sides:

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

Step 3: Simplify the right-hand side

Convert to have a denominator of 36:

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

-\frac{24}{36} + \frac{25}{36} = \frac{1}{36}

Now, we have:

\left(x - \frac{5}{6}\right)^2 = \frac{1}{36}

Step 4: Solve for

Take the square root of both sides:

x - \frac{5}{6} = \pm \frac{\sqrt{1}}{\sqrt{36}}

x - \frac{5}{6} = \pm \frac{1}{6}

Step 5: Solve for

Now, solving for :

1. First root:



x = \frac{5}{6} + \frac{1}{6} = \frac{6}{6} = 1

2. Second root:



x = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}

Final Answer:

x = 1, \quad x = \frac{2}{3} \quad (\text{or } 0.667)

Thus, we successfully solved the equation using the completing the square method!

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