Find the nature of the roots of the following equation, without finding the roots.
`2x^(2)-8x+3=0`

Answer

To find the nature of the roots of the quadratic equation \(2x^2 - 8x + 3 = 0\), 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 equation \(2x^2 - 8x + 3 = 0\), we can identify the coefficients as follows:
- \(a = 2\)
- \(b = -8\)
- \(c = 3\)

### Step 2: Calculate the discriminant
Now, we will substitute these values into the discriminant formula:

\[ D = (-8)^2 - 4 \cdot 2 \cdot 3 \]

Calculating \((-8)^2\):

\[ D = 64 - 4 \cdot 2 \cdot 3 \]

Calculating \(4 \cdot 2 \cdot 3\):

\[ 4 \cdot 2 \cdot 3 = 24 \]

Now substituting this back into the discriminant:

\[ D = 64 - 24 \]

Calculating \(64 - 24\):

\[ D = 40 \]

### Step 3: Determine the nature of the roots
Now that we have the value of the discriminant \(D = 40\), we can determine the nature of the roots:
- If \(D > 0\), the roots are real and unequal.
- If \(D = 0\), the roots are real and equal.
- If \(D < 0\), the roots are complex.

Since \(D = 40\) which is greater than 0, we conclude that the roots of the equation are:

**Real and Unequal.**

### Summary of the Solution
1. Identify coefficients: \(a = 2\), \(b = -8\), \(c = 3\).
2. Calculate discriminant: \(D = 64 - 24 = 40\).
3. Determine nature of roots: Since \(D > 0\), the roots are real and unequal.

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