Solve each of the following equatins :
`3sqrt7x^(2)+4x-sqrt7=0`

Answer

To solve the quadratic equation \(3\sqrt{7}x^{2} + 4x - \sqrt{7} = 0\), we will follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the standard form \(Ax^2 + Bx + C = 0\). Here, we identify:
- \(A = 3\sqrt{7}\)
- \(B = 4\)
- \(C = -\sqrt{7}\)

### Step 2: Calculate the Discriminant
The discriminant \(D\) is given by the formula:
\[ D = B^2 - 4AC \]
Substituting the values of \(A\), \(B\), and \(C\):
\[ D = 4^2 - 4 \times (3\sqrt{7}) \times (-\sqrt{7}) \]
Calculating \(B^2\):
\[ B^2 = 16 \]
Calculating \(4AC\):
\[ 4AC = 4 \times 3\sqrt{7} \times (-\sqrt{7}) = -12 \times 7 = -84 \]
Thus, the discriminant becomes:
\[ D = 16 + 84 = 100 \]

### Step 3: Apply the Quadratic Formula
The solutions for \(x\) are given by:
\[ x = \frac{-B \pm \sqrt{D}}{2A} \]
Substituting the values of \(B\), \(D\), and \(A\):
\[ x = \frac{-4 \pm \sqrt{100}}{2 \times (3\sqrt{7})} \]
Calculating \(\sqrt{100}\):
\[ \sqrt{100} = 10 \]
Thus, we have:
\[ x = \frac{-4 \pm 10}{6\sqrt{7}} \]

### Step 4: Calculate the Two Possible Values of \(x\)
1. For \(x = \frac{-4 + 10}{6\sqrt{7}}\):
\[ x = \frac{6}{6\sqrt{7}} = \frac{1}{\sqrt{7}} \]
Rationalizing the denominator:
\[ x = \frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{7} \]

2. For \(x = \frac{-4 - 10}{6\sqrt{7}}\):
\[ x = \frac{-14}{6\sqrt{7}} = \frac{-7}{3\sqrt{7}} \]
Rationalizing the denominator:
\[ x = \frac{-7}{3\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{-7\sqrt{7}}{21} = \frac{-\sqrt{7}}{3} \]

### Final Solutions
The solutions to the quadratic equation \(3\sqrt{7}x^{2} + 4x - \sqrt{7} = 0\) are:
\[ x = \frac{\sqrt{7}}{7} \quad \text{and} \quad x = \frac{-\sqrt{7}}{3} \]