Solve each of the following quadratic equations:
`4sqrt(6)x^(2)-13x-2sqrt(6)=0`

Explanation

To solve the quadratic equation \( 4\sqrt{6}x^2 - 13x - 2\sqrt{6} = 0 \), we will follow these steps:

### Step 1: Identify the coefficients
The given equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 4\sqrt{6} \)
- \( b = -13 \)
- \( c = -2\sqrt{6} \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-13)^2 - 4(4\sqrt{6})(-2\sqrt{6}) \]
Calculating each part:
\[ D = 169 - 4 \cdot 4 \cdot (-2) \cdot 6 \]
\[ D = 169 + 192 = 361 \]

### Step 3: Find the roots using the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \( b \), \( D \), and \( a \):
\[ x = \frac{-(-13) \pm \sqrt{361}}{2(4\sqrt{6})} \]
\[ x = \frac{13 \pm 19}{8\sqrt{6}} \]

### Step 4: Calculate the two possible values for \( x \)
1. For the positive root:
\[ x_1 = \frac{13 + 19}{8\sqrt{6}} = \frac{32}{8\sqrt{6}} = \frac{4}{\sqrt{6}} = \frac{4\sqrt{6}}{6} = \frac{2\sqrt{6}}{3} \]

2. For the negative root:
\[ x_2 = \frac{13 - 19}{8\sqrt{6}} = \frac{-6}{8\sqrt{6}} = \frac{-3}{4\sqrt{6}} = \frac{-3\sqrt{6}}{24} = \frac{-\sqrt{6}}{8} \]

### Step 5: Final roots
Thus, the roots of the quadratic equation \( 4\sqrt{6}x^2 - 13x - 2\sqrt{6} = 0 \) are:
\[ x_1 = \frac{2\sqrt{6}}{3}, \quad x_2 = \frac{-\sqrt{6}}{8} \]

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