Question

Find the discriminant of the quadratic equation $2 {x}^{2} - 6x + 3 = 0$, and hence find the nature of its roots.

Answer 1

The discriminant of the quadratic equation 2x² - 6x + 3 = 0 is 12, indicating that the equation has two real and distinct roots. The nature of the roots is real and distinct.

Step-by-step explanation:

To find the discriminant of the quadratic equation 2x² - 6x + 3 = 0, we can use the formula: b² - 4ac. In this case, a = 2, b = -6, and c = 3.

So the discriminant is (-6)² - 4(2)(3) = 36 - 24 = 12. Since the discriminant is positive, the quadratic equation has two real and distinct roots. These roots can be found by using the quadratic formula.

The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a). Plugging in the values from the given equation, we have:

x = (-(-6) ± √((-6)² - 4(2)(3))) / (2(2)) = (6 ± √(36 - 24)) / 4 = (6 ± √12) / 4.

Therefore, the nature of the roots is real and distinct.