I've been helping my siblings with their GCSE and A Level maths and I've come across a question where they have just taken the positive square root. It's a pure maths question and there's no (obvious) reason to ignore the negative square root.
I always thought that the square root always gave two values, a positive and negative, and this is shown in the quadratic formula where we have $\pm \sqrt{b^2 - 4ac}$.
So what is the correct answer?
Should you always take just the positive or the negative aswell?
In the same way, if we have $\cos(\theta) = x$, why do we not say then $\theta = \pm \cos^{-1}(x)$ as $\cos(-\theta) = \cos(\theta)$ isn't it?
EDIT: One question wants me to work out the normal of the curve $y^2 = 4ax$ where $a$ is a positive constant at the point $(at^2, 2at)$. They start by taking the square root and write $y = \sqrt{4ax}$.
In another question, they want me to work out the tangent to the line $y^2 = 27x$ at the point $(3,9)$. Here, they also start by taking the square root but write $y = \pm \sqrt{27x}$. Why have they taken the positive in one and both in the other?
It'd probably be easier to do it using implicit differentiation, but I just want to understand why the square root bit is different.