Find the roots of the quadratic equation $3x^2 - 2\sqrt{6}x + 2 = 0$.

Step 1: Identify coefficients

We identify the coefficients $a$, $b$, and $c$ from the given quadratic equation $3x^2 - 2\sqrt{6}x + 2 = 0$ by comparing it with the standard form of a quadratic equation, $ax^2 + bx + c = 0$.

Step 2: Calculate the Discriminant

First, we calculate the discriminant $D$ using the formula $D = b^2 - 4ac$. This value will determine the nature of the roots. Here, $a=3$, $b=-2\sqrt{6}$, and $c=2$. We substitute these values into the formula.

Step 3: Determine the nature of roots

Since the discriminant $D = 0$, the quadratic equation has two equal real roots. This is an important property that simplifies finding the roots, as mentioned in the NCERT textbook (reference 1, 3).

Step 4: Apply the quadratic formula

To find the roots, we use the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Since we already calculated $b^2 - 4ac = 0$, this simplifies the formula significantly.

Step 5: Substitute values and simplify

We substitute the values of $a$, $b$, and the discriminant $D=0$ into the quadratic formula. This leads to a single value for $x$, representing the two equal real roots. We then simplify the expression to get the final root.

Step 6: Express the roots

Since the discriminant was zero, the quadratic equation has two equal real roots. Therefore, both roots are $\frac{\sqrt{6}}{3}$.

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