$ ext{The roots of the quadratic equation } 3x^2 - 2\sqrt{6}x + 2 = 0 ext{ are: }$

Q: $\text{The roots of the quadratic equation } 3x^2 - 2\sqrt{6}x + 2 = 0 \text{ are: }$

To determine the roots of the quadratic equation $3x^2 - 2\sqrt{6}x + 2 = 0$, we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where $a = 3$, $b = -2\sqrt{6}$, and $c = 2$. First, calculate the discriminant ($b^2 - 4ac$): \[ b^2 = (-2\sqrt{6})^2 = 4 \times 6 = 24 \] \[ 4ac = 4 \times 3 \times 2 = 24 \] \[ b^2 - 4ac = 24 - 24 = 0 \] Since the discriminant is zero, the quadratic equation has exactly one unique root, which is a repeated root. Now, substitute the values into the quadratic formula: \[ x = \frac{-(-2\sqrt{6}) \pm \sqrt{0}}{2 \times 3} = \frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3} \] Simplifying $\frac{\sqrt{6}}{3}$, we get: \[ x = \sqrt{\frac{2}{3}} \] Thus, the roots of the equation are $\sqrt{\frac{2}{3}}$ and $\sqrt{\frac{2}{3}}$, which confirms that Option 1 is correct. **Explanation for why other options are incorrect:** - **Option 2: $\sqrt{\frac{3}{2}}, \sqrt{\frac{3}{2}}$** This option suggests roots that do not match the calculated root $\sqrt{\frac{2}{3}}$. The calculation clearly shows that the correct root is $\sqrt{\frac{2}{3}}$, not $\sqrt{\frac{3}{2}}$. - **Option 3: 1, 2** These are integer values and do not match the calculated root $\sqrt{\frac{2}{3}}$. The quadratic equation does not yield integer roots. - **Option 4: 2, 3** Similar to Option 3, these are integer values and do not match the calculated root $\sqrt{\frac{2}{3}}$. The equation does not produce these roots. In conclusion, the correct roots of the quadratic equation are $\sqrt{\frac{2}{3}}$ and $\sqrt{\frac{2}{3}}$, which supports Option 1 as the correct answer.

Question

$	ext{The roots of the quadratic equation } 3x^2 - 2\sqrt{6}x + 2 = 0 	ext{ are: }$

Accepted Answer

To determine the roots of the quadratic equation $3x^2 - 2\sqrt{6}x + 2 = 0$, we can use the quadratic formula:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

where $a = 3$, $b = -2\sqrt{6}$, and $c = 2$.

First, calculate the discriminant ($b^2 - 4ac$):

$$
b^2 = (-2\sqrt{6})^2 = 4 \times 6 = 24
$$

$$
4ac = 4 \times 3 \times 2 = 24
$$

$$
b^2 - 4ac = 24 - 24 = 0
$$

Since the discriminant is zero, the quadratic equation has exactly one unique root, which is a repeated root. Now, substitute the values into the quadratic formula:

$$
x = \frac{-(-2\sqrt{6}) \pm \sqrt{0}}{2 \times 3} = \frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3}
$$

Simplifying $\frac{\sqrt{6}}{3}$, we get:

$$
x = \sqrt{\frac{2}{3}}
$$

Thus, the roots of the equation are $\sqrt{\frac{2}{3}}$ and $\sqrt{\frac{2}{3}}$, which confirms that Option 1 is correct.

**Explanation for why other options are incorrect:**

- **Option 2: $\sqrt{\frac{3}{2}}, \sqrt{\frac{3}{2}}$**

This option suggests roots that do not match the calculated root $\sqrt{\frac{2}{3}}$. The calculation clearly shows that the correct root is $\sqrt{\frac{2}{3}}$, not $\sqrt{\frac{3}{2}}$.

- **Option 3: 1, 2**

These are integer values and do not match the calculated root $\sqrt{\frac{2}{3}}$. The quadratic equation does not yield integer roots.

- **Option 4: 2, 3**

Similar to Option 3, these are integer values and do not match the calculated root $\sqrt{\frac{2}{3}}$. The equation does not produce these roots.

In conclusion, the correct roots of the quadratic equation are $\sqrt{\frac{2}{3}}$ and $\sqrt{\frac{2}{3}}$, which supports Option 1 as the correct answer.