Find the roots of the equations.
Q. `5x^(2)-x+4=0`.

Answer

To find the roots of the quadratic equation \(5x^2 - x + 4 = 0\), we will follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\).
Here, we have:
- \(a = 5\)
- \(b = -1\)
- \(c = 4\)

### Step 2: Calculate the discriminant
The discriminant \(D\) is calculated using the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-1)^2 - 4 \cdot 5 \cdot 4 \]
Calculating this gives:
\[ D = 1 - 80 = -79 \]

### Step 3: Analyze the discriminant
Since the discriminant \(D\) is less than 0 (\(D < 0\)), this indicates that the roots of the equation are imaginary (or complex).

### Step 4: Use 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 \(a\):
\[ x = \frac{-(-1) \pm \sqrt{-79}}{2 \cdot 5} \]
This simplifies to:
\[ x = \frac{1 \pm \sqrt{-79}}{10} \]

### Step 5: Express the square root of the negative number
Since \(\sqrt{-79} = i\sqrt{79}\), where \(i\) is the imaginary unit, we can rewrite the expression for \(x\):
\[ x = \frac{1 \pm i\sqrt{79}}{10} \]

### Step 6: Write the final roots
Thus, the two roots of the equation are:
\[ x_1 = \frac{1}{10} + \frac{\sqrt{79}}{10}i \]
\[ x_2 = \frac{1}{10} - \frac{\sqrt{79}}{10}i \]

### Summary of Roots
The roots of the equation \(5x^2 - x + 4 = 0\) are:
- \(x_1 = \frac{1}{10} + \frac{\sqrt{79}}{10}i\)
- \(x_2 = \frac{1}{10} - \frac{\sqrt{79}}{10}i\)

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