Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
`2x^(2)+6sqrt(3)x-60=0`

Answer

To find the roots of the quadratic equation \(2x^2 + 6\sqrt{3}x - 60 = 0\) using the quadratic formula, we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Here, we can identify:
- \(a = 2\)
- \(b = 6\sqrt{3}\)
- \(c = -60\)

### Step 2: Calculate the discriminant
The discriminant \(D\) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (6\sqrt{3})^2 - 4 \cdot 2 \cdot (-60) \]
Calculating \(b^2\):
\[ (6\sqrt{3})^2 = 36 \cdot 3 = 108 \]
Calculating \(4ac\):
\[ 4 \cdot 2 \cdot (-60) = -480 \quad \text{(since 4 * 2 = 8 and 8 * -60 = -480)} \]
Now substituting back into the discriminant:
\[ D = 108 - (-480) = 108 + 480 = 588 \]

### Step 3: Apply the quadratic formula
The quadratic formula to find the roots is:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \(b\), \(D\), and \(a\):
\[ x = \frac{-6\sqrt{3} \pm \sqrt{588}}{2 \cdot 2} \]
Calculating \(2a\):
\[ 2a = 2 \cdot 2 = 4 \]
Now we need to simplify \(\sqrt{588}\):
\[ \sqrt{588} = \sqrt{196 \cdot 3} = 14\sqrt{3} \]
Now substituting this back into the formula:
\[ x = \frac{-6\sqrt{3} \pm 14\sqrt{3}}{4} \]

### Step 4: Simplify the expression
This gives us two cases for \(x\):
1. \(x = \frac{-6\sqrt{3} + 14\sqrt{3}}{4}\)
2. \(x = \frac{-6\sqrt{3} - 14\sqrt{3}}{4}\)

Calculating the first case:
\[ x = \frac{8\sqrt{3}}{4} = 2\sqrt{3} \]

Calculating the second case:
\[ x = \frac{-20\sqrt{3}}{4} = -5\sqrt{3} \]

### Final Roots
Thus, the roots of the equation \(2x^2 + 6\sqrt{3}x - 60 = 0\) are:
\[ x = 2\sqrt{3} \quad \text{and} \quad x = -5\sqrt{3} \]

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