Solve the following quations by using qardratic formula:
`5x^(2)-19x+17=0`

Answer

To solve the quadratic equation \(5x^2 - 19x + 17 = 0\) using the quadratic formula, we will follow these steps:

### Step 1: Identify the coefficients
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Here, we can identify:
- \(a = 5\)
- \(b = -19\)
- \(c = 17\)

### Step 2: Write down the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Calculate \(b^2 - 4ac\)
Now, we need to calculate the discriminant \(b^2 - 4ac\):
\[ b^2 = (-19)^2 = 361 \]
\[ 4ac = 4 \times 5 \times 17 = 340 \]
Now, substitute these values into the discriminant:
\[ b^2 - 4ac = 361 - 340 = 21 \]

### Step 4: Substitute values into the quadratic formula
Now we can substitute \(a\), \(b\), and the discriminant into the quadratic formula:
\[ x = \frac{-(-19) \pm \sqrt{21}}{2 \times 5} \]
This simplifies to:
\[ x = \frac{19 \pm \sqrt{21}}{10} \]

### Step 5: Write the final solutions
Thus, the solutions for the equation \(5x^2 - 19x + 17 = 0\) are:
\[ x = \frac{19 + \sqrt{21}}{10} \quad \text{and} \quad x = \frac{19 - \sqrt{21}}{10} \]

### Summary of the solutions
The roots of the equation are:
1. \(x_1 = \frac{19 + \sqrt{21}}{10}\)
2. \(x_2 = \frac{19 - \sqrt{21}}{10}\)

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