Quadratic Equation: 2x^2 - 4x + 3 = 0

Determining the Nature of Roots:

To determine the nature of the roots of a quadratic equation, we can use the discriminant, which is the expression under the square root sign in the quadratic formula.

The quadratic formula is given as:
x = (-b ± √(b^2 - 4ac)) / 2a

The discriminant (D) is given as:
D = b^2 - 4ac

Discriminant Calculation:

In the given equation, a = 2, b = -4, and c = 3. Let's calculate the discriminant.

D = (-4)^2 - 4(2)(3)
D = 16 - 24
D = -8

Nature of Roots:

The nature of roots can be determined based on the value of the discriminant (D).

1. If D > 0 (positive discriminant):
- The quadratic equation has two distinct real roots.
- The graph of the equation will intersect the x-axis at two different points.

2. If D = 0 (zero discriminant):
- The quadratic equation has one real root.
- The graph of the equation will touch the x-axis at one point (tangent).

3. If D < 0="" (negative="" />
- The quadratic equation has two complex roots (conjugate pairs).
- The graph of the equation will not intersect the x-axis, as the roots are imaginary.

Conclusion:

In the given quadratic equation 2x^2 - 4x + 3 = 0, the discriminant (D) is -8, which is negative. Therefore, the equation has two complex roots, indicating that it does not intersect the x-axis.

1) First of all please ask the question properly some signs are missing.