Solve the following quadratic: `\ 5x^2-6x-2=0`

Explanation

To solve the quadratic equation \( 5x^2 - 6x - 2 = 0 \), 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:
- \( a = 5 \)
- \( b = -6 \)
- \( c = -2 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-6)^2 - 4 \cdot 5 \cdot (-2) \]
Calculating this:
\[ D = 36 + 40 = 76 \]

### Step 3: Determine the nature of the roots
Since the discriminant \( D = 76 \) is positive, this indicates that the quadratic equation has two distinct real roots.

### Step 4: Use the quadratic formula to find the roots
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \( b \), \( D \), and \( a \):
\[ x = \frac{-(-6) \pm \sqrt{76}}{2 \cdot 5} \]
This simplifies to:
\[ x = \frac{6 \pm \sqrt{76}}{10} \]

### Step 5: Simplify \( \sqrt{76} \)
We can simplify \( \sqrt{76} \):
\[ \sqrt{76} = \sqrt{4 \cdot 19} = 2\sqrt{19} \]
Thus, substituting back:
\[ x = \frac{6 \pm 2\sqrt{19}}{10} \]
This can be simplified further:
\[ x = \frac{6}{10} \pm \frac{2\sqrt{19}}{10} \]
\[ x = \frac{3}{5} \pm \frac{\sqrt{19}}{5} \]

### Step 6: Write the final roots
The roots of the equation \( 5x^2 - 6x - 2 = 0 \) are:
\[ x_1 = \frac{3 + \sqrt{19}}{5}, \quad x_2 = \frac{3 - \sqrt{19}}{5} \]

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