What is the nature of the roots of $$3x^2+6x-5=0$$​ ?

The quadratic equation is $$3x^2+6x − 5=0$$​.

To determine the nature of the roots, we calculate the discriminant (D) of the quadratic equation. The discriminant is given by $$D=b^2−4ac$$​. The nature of the roots depends on the value of D:

If D > 0, the roots are real and distinct.

If D = 0, the roots are real and equal.

If D < 0, the roots are complex (non-real) and conjugate.

From $$3x^2+6x−5=0:$$​​

a = 3, b = 6, c = −5

Now,

D =$$b^2−4ac$$​​

D = $$62−4⋅3⋅(−5)$$​​

D = 36 + 60 = 96

Since D = 96 > 0, the roots are real and distinct.
Thus, the roots of the equation 3x^2+6x−5 = 0 are real and distinct.