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

To determine the nature of the roots of the quadratic equation, it is essential to calculate the discriminant without solving for the actual roots. Let's consider the given equation:

$$2x^2 - 8x + 3 = 0$$

A quadratic equation can be generalized in the form:

$$ax^2 + bx + c = 0$$

By comparing this general equation with the given equation, we get:

• $a = 2$
• $b = -8$
• $c = 3$

The discriminant ($D$) of a quadratic equation is given by:

$$D = b^2 - 4ac$$

Substitute the values of $a$, $b$, and $c$ into the discriminant formula:

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

Let's calculate this step-by-step:

1. Compute $(-8)^2$:

$$(-8)^2 = 64$$

2. Compute $4 \cdot 2 \cdot 3$:

$$4 \cdot 2 \cdot 3 = 24$$

3. Subtract the second result from the first:

$$64 - 24 = 40$$

Thus, the discriminant $D$ is:

$$D = 40$$

Since the discriminant $D$ is greater than 0 ($40 > 0$), this indicates that the equation has:

• Two real roots
• The roots are distinct and unequal

Moreover, because the discriminant is not a perfect square (40 is not a perfect square), the roots will be irrational and will form a pair of conjugate surds.

Conclusion:

The nature of the roots of the quadratic equation $2x^2 - 8x + 3 = 0$ are real, distinct, and irrational.