The question is:
$x^2 +2(a-1)x + a + 5 = 0$ For what value of $a$ will the roots of the quadratic equation be equal in magnitude and opposite in sign?
For a general quadratic $ax^2+bx+c = 0$, the sum of roots is equal to $\frac{-b}{a}$. So we just need to set that to zero here. $$\frac{-2(a-1)}{1}= 0$$
$$\therefore a = 1$$
But if we put $a = 1$ into the original question, our equation becomes $x^2+6=0$ which has no solutions. So the answer for this question is $\phi$. My question is, how am I supposed to know when to test my solutions by plugging them into the original question? Normally when dealing with radicals, inequalities, modulus and when squaring both sides is involved, it's common to have to do this. Here I see none of those things. If anyone can give me a thumb rule for when I should check my solutions and potentially reject some, that would be great!