How to find the positive and negative roots of a function?

Question

Iam trying to solve the following question:

Find all numbers $a$, such that the equation $x^2-ax-a = 0$ has one positive root and one negative root.

I've tried it already but I cannot seem to understand what the question is asking for, so how would I go about solving it? Also how do I find the positive and negative roots of a function in general?

Answer 1

Use the quadratic formula to solve for $x$. Use what is inside the square root to find the values of $a$ that give two values for $x$. (The contents of the square root, which is an expression in $a$, must be positive.)

Then for the value of $x$ that comes from subtracting the square root, solve the inequality that makes that negative. For the value of $x$ that comes from adding the square root, solve the inequality that makes that positive.

You now have three sets for the values of $a$. The one you want is the intersection of all three of those sets.

You do not need to "find the positive and negative roots of a function in general," just for this particular problem that uses a quadratic equation. That is much easier than the general problem.

Answer 2

You could use the fact that the product of the two roots is $-a$.

If they have different signs, their product has to be negative. That means $-a<0$.

Also two distinct roots means the discriminant is greater than $0$. This gives you $a^2+4a>0$.

Combining the two should give you the answer.