Solve each of the following equatins :
`2sqrt5x^(2)-3x-sqrt5=0`

Answer

To solve the quadratic equation \( 2\sqrt{5}x^2 - 3x - \sqrt{5} = 0 \), we will follow these steps:

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

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-3)^2 - 4 \cdot (2\sqrt{5}) \cdot (-\sqrt{5}) \]
Calculating each term:
\[ D = 9 - 4 \cdot 2\sqrt{5} \cdot (-\sqrt{5}) = 9 + 8 \cdot 5 = 9 + 40 = 49 \]

### Step 3: Find the roots using the quadratic formula
The roots of the quadratic equation can be calculated using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \( b \), \( D \), and \( a \):
\[ x = \frac{-(-3) \pm \sqrt{49}}{2 \cdot (2\sqrt{5})} \]
This simplifies to:
\[ x = \frac{3 \pm 7}{4\sqrt{5}} \]

### Step 4: Calculate the two possible values for \( x \)
1. First root:
\[ x_1 = \frac{3 + 7}{4\sqrt{5}} = \frac{10}{4\sqrt{5}} = \frac{5}{2\sqrt{5}} = \frac{\sqrt{5} \cdot 5}{2 \cdot 5} = \frac{\sqrt{5}}{2} \]

2. Second root:
\[ x_2 = \frac{3 - 7}{4\sqrt{5}} = \frac{-4}{4\sqrt{5}} = \frac{-1}{\sqrt{5}} \]

### Final Solutions
The solutions to the quadratic equation \( 2\sqrt{5}x^2 - 3x - \sqrt{5} = 0 \) are:
\[ x_1 = \frac{\sqrt{5}}{2} \quad \text{and} \quad x_2 = -\frac{1}{\sqrt{5}} \]

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