Find the nature of roots of the quadratic equation `4x^(2)-5x+3=0`.

Answer

To find the nature of the roots of the quadratic equation \(4x^2 - 5x + 3 = 0\), we will use the discriminant method. The discriminant \(D\) is given by the formula:

\[ D = b^2 - 4ac \]

### Step 1: Identify the coefficients
In the given quadratic equation \(4x^2 - 5x + 3 = 0\), we can identify the coefficients:
- \(a = 4\)
- \(b = -5\)
- \(c = 3\)

### Step 2: Substitute the coefficients into the discriminant formula
Now, we will substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:

\[ D = (-5)^2 - 4 \cdot 4 \cdot 3 \]

### Step 3: Calculate \(b^2\)
Calculating \(b^2\):

\[ D = 25 - 4 \cdot 4 \cdot 3 \]

### Step 4: Calculate \(4ac\)
Now calculate \(4ac\):

\[ 4 \cdot 4 \cdot 3 = 48 \]

### Step 5: Substitute back into the discriminant
Now substitute this back into the discriminant calculation:

\[ D = 25 - 48 \]

### Step 6: Calculate the discriminant
Now, perform the subtraction:

\[ D = 25 - 48 = -23 \]

### Step 7: Determine the nature of the roots
Since the discriminant \(D\) is less than zero (\(D < 0\)), we conclude that the quadratic equation has no real roots. Therefore, the nature of the roots is:

- The equation has complex (imaginary) roots.

### Final Conclusion
The nature of the roots of the quadratic equation \(4x^2 - 5x + 3 = 0\) is that it has no real roots.

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