Finding roots of polynomials, negative square root The formula for finding the roots of a polynomial is as follows
$$x = \frac {-b \pm \sqrt{ b^2 - 4ac }}{2a} $$

what happens if you want to find the roots of a polynomial like this simplified one

$$ 3x^2 + x + 24 = 0 $$

then the square root value becomes

$$ \sqrt{ 1^2 - 4\cdot3\cdot24 } $$


$$ = \sqrt{ -287 } $$
which is the square root of a negative number, which isn't allowed. What do you do in this case? I know there are other methods, i.e. factorisation and completing the square, but does this mean that this formula can only be used in specialised cases or have i gone wrong somewhere along the path?

100% correct, and good observation.
To solve this, we define $\sqrt{-1}=i$ where $i$ is the imaginary unit
Then $\sqrt{-287}=\sqrt{287}i$, and we can solve as per the general quadratic formula. Numbers of the form $a+bi$ are known as complex numbers and are extremely useful.
In general the term $b^2-4ac$ is known as the discriminant of the quadratic equation. It should be clear that if $b^2-4ac>0$ there exists two real solutions, if $b^2-4ac=0$ there is one solution (the repeated root) and if 
$$b^2-4ac \lt 0$$ there are two complex solutions.
The quadratic formula is the most general way to solve the quadratic equation - so you are doing the right thing.
