Solve each of the following quadratic equations:
`sqrt(7)x^(2)-6x-13sqrt(7)=0`

Explanation

To solve the quadratic equation \(\sqrt{7}x^2 - 6x - 13\sqrt{7} = 0\), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be rewritten in the standard form \(ax^2 + bx + c = 0\). Here:
- \(a = \sqrt{7}\)
- \(b = -6\)
- \(c = -13\sqrt{7}\)

### Step 2: Calculate the discriminant
The discriminant \(D\) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-6)^2 - 4(\sqrt{7})(-13\sqrt{7}) \]
Calculating this:
\[ D = 36 - 4 \cdot \sqrt{7} \cdot -13\sqrt{7} \]
\[ D = 36 + 4 \cdot 13 \cdot 7 \]
\[ D = 36 + 364 = 400 \]

### 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 \(a\):
\[ x = \frac{-(-6) \pm \sqrt{400}}{2\sqrt{7}} \]
\[ x = \frac{6 \pm 20}{2\sqrt{7}} \]

### Step 4: Calculate the two possible values for \(x\)
1. First root:
\[ x_1 = \frac{6 + 20}{2\sqrt{7}} = \frac{26}{2\sqrt{7}} = \frac{13}{\sqrt{7}} \]

2. Second root:
\[ x_2 = \frac{6 - 20}{2\sqrt{7}} = \frac{-14}{2\sqrt{7}} = \frac{-7}{\sqrt{7}} = -\sqrt{7} \]

### Final Solution
The roots of the quadratic equation \(\sqrt{7}x^2 - 6x - 13\sqrt{7} = 0\) are:
\[ x = \frac{13}{\sqrt{7}} \quad \text{and} \quad x = -\sqrt{7} \]

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