How do you find all the real and complex roots of #12x²+8x-15=0#?

1 Answer

Answer:

The equation has two rational roots #5/6# and #-3/2#.

Explanation:

Comparing the equation #12x²+8x−15=0#, with general form of a quadratic equation i.e. #ax²+bx+c=0#, we observe that #a=12, b=8 and c--15#. Note that a quadratic equation in one variable will have two roots.

Hence discriminant #b^2-4ac# equals #8^2-4*12*(-15)# or #64+720# ie. #784#.

As in this equation discriminant #b^2-4ac>=0#, the roots are real and as #sqrt784=28#, roots are rational.

Roots of a general quadratic equation #ax²+bx+c=0# are #(-b+-sqrt(b^2-4a))/2a#. Hence the roots are

#(-8+-28)/24# i.e. #5/6# and #-3/2#.