Question

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

Answer 1

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`.