The roots of the equation ` 6 sqrt(5) x^(2) - 9x - 3sqrt(5) = 0` is

Answer

To find the roots of the quadratic equation \( 6\sqrt{5}x^2 - 9x - 3\sqrt{5} = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 6\sqrt{5} \)
- \( b = -9 \)
- \( c = -3\sqrt{5} \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-9)^2 - 4(6\sqrt{5})(-3\sqrt{5}) \]
Calculating each term:
\[ D = 81 - 4 \cdot 6\sqrt{5} \cdot (-3\sqrt{5}) \]
\[ D = 81 + 4 \cdot 6 \cdot 3 \cdot 5 \]
\[ D = 81 + 72 = 153 \]

### 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 \( a \):
\[ x = \frac{-(-9) \pm \sqrt{153}}{2(6\sqrt{5})} \]
\[ x = \frac{9 \pm \sqrt{153}}{12\sqrt{5}} \]

### Step 4: Simplify the roots
Now we can simplify \( \sqrt{153} \):
\[ \sqrt{153} = \sqrt{9 \cdot 17} = 3\sqrt{17} \]
Substituting this back into the equation for \( x \):
\[ x = \frac{9 \pm 3\sqrt{17}}{12\sqrt{5}} \]
This can be further simplified:
\[ x = \frac{3(3 \pm \sqrt{17})}{12\sqrt{5}} = \frac{3 \pm \sqrt{17}}{4\sqrt{5}} \]

### Final Roots
Thus, the roots of the equation \( 6\sqrt{5}x^2 - 9x - 3\sqrt{5} = 0 \) are:
\[ x_1 = \frac{3 + \sqrt{17}}{4\sqrt{5}}, \quad x_2 = \frac{3 - \sqrt{17}}{4\sqrt{5}} \]

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