Solve the following quations by using qardratic formula:
`sqrt6x^(2)-4x-2sqrt6=0`

Answer

To solve the quadratic equation \( \sqrt{6}x^2 - 4x - 2\sqrt{6} = 0 \) using the quadratic formula, we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). From the given equation, we can identify:
- \( a = \sqrt{6} \)
- \( b = -4 \)
- \( c = -2\sqrt{6} \)

### Step 2: Write the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Calculate \( b^2 - 4ac \)
Now, we need to calculate \( b^2 - 4ac \):
\[ b^2 = (-4)^2 = 16 \]
\[ 4ac = 4 \cdot \sqrt{6} \cdot (-2\sqrt{6}) = -8 \cdot 6 = -48 \]
Now, substituting these values:
\[ b^2 - 4ac = 16 - (-48) = 16 + 48 = 64 \]

### Step 4: Substitute into the quadratic formula
Now, substitute \( b \), \( b^2 - 4ac \), and \( a \) into the quadratic formula:
\[ x = \frac{-(-4) \pm \sqrt{64}}{2\sqrt{6}} = \frac{4 \pm 8}{2\sqrt{6}} \]

### Step 5: Calculate the two possible values for \( x \)
Now we will calculate the two possible values for \( x \):
1. \( x_1 = \frac{4 + 8}{2\sqrt{6}} = \frac{12}{2\sqrt{6}} = \frac{6}{\sqrt{6}} = \sqrt{6} \)
2. \( x_2 = \frac{4 - 8}{2\sqrt{6}} = \frac{-4}{2\sqrt{6}} = \frac{-2}{\sqrt{6}} \)

### Step 6: Rationalize the denominator for \( x_2 \)
To rationalize \( x_2 \):
\[ x_2 = \frac{-2}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{-2\sqrt{6}}{6} = \frac{-\sqrt{6}}{3} \]

### Final Solutions
Thus, the solutions to the equation \( \sqrt{6}x^2 - 4x - 2\sqrt{6} = 0 \) are:
\[ x_1 = \sqrt{6} \quad \text{and} \quad x_2 = -\frac{\sqrt{6}}{3} \]